Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
The number of facets of three-dimensional Dirichlet stereohedra
Francisco Santos (w. D. Bochis, P. Sabariego), U. Cantabria.
ERC Workshop “Delaunay Geometry”; Berlin, oct 2013
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The question How many facets can a 3-d Dirichlet stereohedron have?
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The question How many facets can a 3-d Dirichlet stereohedron have?
Stereohedron: polytope that tiles Rn (face-to-face) by the action of a group of motions (a crystallographic space group G).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The question How many facets can a 3-d Dirichlet stereohedron have?
Stereohedron: polytope that tiles Rn (face-to-face) by the action of a group of motions (a crystallographic space group G). Dirichlet stereohedron: the Voronoi region of a point p ∈ S with respect to an orbit S of G.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The question How many facets can a 3-d Dirichlet stereohedron have?
Stereohedron: polytope that tiles Rn (face-to-face) by the action of a group of motions (a crystallographic space group G). Dirichlet stereohedron: the Voronoi region of a point p ∈ S with respect to an orbit S of G.
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F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The question How many facets can a 3-d Dirichlet stereohedron have?
Stereohedron: polytope that tiles Rn (face-to-face) by the action of a group of motions (a crystallographic space group G). Dirichlet stereohedron: the Voronoi region of a point p ∈ S with respect to an orbit S of G.
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F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History
Somehow related to Hilbert’s XVIII problem.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History
Somehow related to Hilbert’s XVIII problem. Fedorov (1885) classified 3-d parallelohedra. In particular, found (Dirichlet) stereohedra with 14 facets.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History
Somehow related to Hilbert’s XVIII problem. Fedorov (1885) classified 3-d parallelohedra. In particular, found (Dirichlet) stereohedra with 14 facets. F¨oppl (1916), Novacki (1935), Smith (1965), Stogrin (1968), Koch (1972), Koch-Fisher (1974) found (Dirichlet) stereohedra with 16, 18, 20, 23 and 24 facets.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History
Somehow related to Hilbert’s XVIII problem. Fedorov (1885) classified 3-d parallelohedra. In particular, found (Dirichlet) stereohedra with 14 facets. F¨oppl (1916), Novacki (1935), Smith (1965), Stogrin (1968), Koch (1972), Koch-Fisher (1974) found (Dirichlet) stereohedra with 16, 18, 20, 23 and 24 facets. Delone (1961) proved that no 3-d stereohedron can have more than 390 facets.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History
Somehow related to Hilbert’s XVIII problem. Fedorov (1885) classified 3-d parallelohedra. In particular, found (Dirichlet) stereohedra with 14 facets. F¨oppl (1916), Novacki (1935), Smith (1965), Stogrin (1968), Koch (1972), Koch-Fisher (1974) found (Dirichlet) stereohedra with 16, 18, 20, 23 and 24 facets. Delone (1961) proved that no 3-d stereohedron can have more than 390 facets. Engel (1980) found Dirichlet stereohedra with 38 facets for the cubic group I 4132
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History
Somehow related to Hilbert’s XVIII problem. Fedorov (1885) classified 3-d parallelohedra. In particular, found (Dirichlet) stereohedra with 14 facets. F¨oppl (1916), Novacki (1935), Smith (1965), Stogrin (1968), Koch (1972), Koch-Fisher (1974) found (Dirichlet) stereohedra with 16, 18, 20, 23 and 24 facets. Delone (1961) proved that no 3-d stereohedron can have more than 390 facets. Engel (1980) found Dirichlet stereohedra with 38 facets for the cubic group I 4132 Our goal: lower the gap between 38 and 390 (by improving the upper bound).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. History
Somehow related to Hilbert’s XVIII problem. Fedorov (1885) classified 3-d parallelohedra. In particular, found (Dirichlet) stereohedra with 14 facets. F¨oppl (1916), Novacki (1935), Smith (1965), Stogrin (1968), Koch (1972), Koch-Fisher (1974) found (Dirichlet) stereohedra with 16, 18, 20, 23 and 24 facets. Delone (1961) proved that no 3-d stereohedron can have more than 390 facets. Engel (1980) found Dirichlet stereohedra with 38 facets for the cubic group I 4132 Our goal: lower the gap between 38 and 390 (by improving the upper bound). Our global upper bound: 92.
F. Santos Number of facets of 3-d Dirichlet stereohedra W 967 a va a o x. ov, a o o, a g o a yg g av 1, 2 o 3 va a aa, a a o o , o v a o o o , va. y g a o a a o vaay o aa v o go a a, a a go o o o a y o . a o g 12 a 13, a o a o o go y ao a a o a o a . aga x a gv a ay o, a a, vy o o a ay y o o o o va o . a o o o a o a ov a oo o og o o a vy Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups a o oga o. o a intro. a o o aa, v gao, a v Engel’s o stereohedra, a a o xy .
F. Santos Number of facets of 3-d Dirichlet stereohedra
13 a 13. x a o y o oa o y g 1980. g ay v gao o o y o aa o o a, g a o o aa va « 0.0158. a a o aa va 0.20 x 0.23 a 0.14 0.20, o g a o a ag a 0.214 x 0.220, 0.145 . y o — p, a o a go o a, a o , o v, a p a a o g y a o a Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Why Dirichlet stereohedra?
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Why Dirichlet stereohedra?
“The total absence of methods usable in more general situations” (Gr¨unbaum-Shephard, 1980).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Why Dirichlet stereohedra?
“The total absence of methods usable in more general situations” (Gr¨unbaum-Shephard, 1980). Even if “there seems to be no grounds to assume that all stereohedra are combinatorially equivalent to Dirichlet stereohedra” (op. cit.)
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Why Dirichlet stereohedra?
“The total absence of methods usable in more general situations” (Gr¨unbaum-Shephard, 1980). Even if “there seems to be no grounds to assume that all stereohedra are combinatorially equivalent to Dirichlet stereohedra” (op. cit.) Also, there is an algorithm to compute all Dirichlet stereohedra in a given dimension “if carried out with sufficient perseverance”, (op. cit.). The same is not true for arbitrary stereohedra.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The zoo
Our method combines general principles with case-by-case studies.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The zoo
Our method combines general principles with case-by-case studies. In particular, we start by dividing the 219 types of 3-d crystallographic groups into three blocks:
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The zoo
Our method combines general principles with case-by-case studies. In particular, we start by dividing the 219 types of 3-d crystallographic groups into three blocks: Groups that contain reflection planes (the reptilarium).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The zoo
Our method combines general principles with case-by-case studies. In particular, we start by dividing the 219 types of 3-d crystallographic groups into three blocks: Groups that contain reflection planes (the reptilarium). Rest of non-cubic groups (fish and birds).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The zoo
Our method combines general principles with case-by-case studies. In particular, we start by dividing the 219 types of 3-d crystallographic groups into three blocks: Groups that contain reflection planes (the reptilarium). Rest of non-cubic groups (fish and birds). Rest of cubic groups (mammals),
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. The zoo
Our method combines general principles with case-by-case studies. In particular, we start by dividing the 219 types of 3-d crystallographic groups into three blocks: Groups that contain reflection planes (the reptilarium). Rest of non-cubic groups (fish and birds). Rest of cubic groups (mammals), divided into “full cubic groups” (the petting zoo) and “quarter cubic groups” (the wild beasts).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Summary of results
nbr. of our upper biggest nbr. of “wild” groups bound example groups (bd > 38) “3” reflections 28 8 8 – “2” reflections 40 18 18 – “1” reflection 32 15 15 – Non-cubic 97 80 32 21 Cubic, full 14 25 17 – Cubic,quarter 8 92 38 8 TOTAL 219 92 38 29
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Crash-course on 3-d crystallographic groups
Bieberbach: A discrete group of motions in E d is crystallographic iff it contains d independent translations (a full-dimensional lattice). Two such groups are affinely equivalent if and only if they are isomorphic.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Crash-course on 3-d crystallographic groups
Bieberbach: A discrete group of motions in E d is crystallographic iff it contains d independent translations (a full-dimensional lattice). Two such groups are affinely equivalent if and only if they are isomorphic. Crystallographic groups are primarily classified by their lattice type (14 Bravais types) and their point group (quotient by translation subgroup. There are 32 point groups (the discrete subgroups of O(3) satisfying the “crystallographic restriction”).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Aerial view, and some animals
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Aerial view, and some animals
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Aerial view, and some animals
1/2
1/2 1/2 1/2 1/2 1/2
1/2 1/2 1/2 1/2 1/2 1/2
42 P n P4222 P42212
4 422 m
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Delone’s bound
Let us call aspects of a crystallographic group G the elements of its point group. Fundamental theorem of stereohedra (Delone 1961) Stereohedra for a d-dimensional crystallographic group with a aspects cannot have more than 2d (a + 1) − 2 facets.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Delone’s bound
Let us call aspects of a crystallographic group G the elements of its point group. Fundamental theorem of stereohedra (Delone 1961) Stereohedra for a d-dimensional crystallographic group with a aspects cannot have more than 2d (a + 1) − 2 facets.
3-d crystallographic groups have a maximum of 48 aspects (symmetries of the 3-cube).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Delone’s bound
Let us call aspects of a crystallographic group G the elements of its point group. Fundamental theorem of stereohedra (Delone 1961) Stereohedra for a d-dimensional crystallographic group with a aspects cannot have more than 2d (a + 1) − 2 facets.
3-d crystallographic groups have a maximum of 48 aspects (symmetries of the 3-cube). Non-cubic groups can have up to 24 aspects (hexagonal prism).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Proof of Delone’s bound (for Dirichlet stereohedra)
Let p be a base point for an orbit Gp, and consider separately its a translational orbits. That is, let Gp = O1 ∪ O2 ∪ ∪ Oa, with p ∈ O1.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Proof of Delone’s bound (for Dirichlet stereohedra)
Let p be a base point for an orbit Gp, and consider separately its a translational orbits. That is, let Gp = O1 ∪ O2 ∪ ∪ Oa, with ′ p ∈ O1. Let p ∈ Gp \ p. Then, a necessary condition for VorGp(p) ′ and VorGp(p ) to have a common facet is:
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Proof of Delone’s bound (for Dirichlet stereohedra)
Let p be a base point for an orbit Gp, and consider separately its a translational orbits. That is, let Gp = O1 ∪ O2 ∪ ∪ Oa, with ′ p ∈ O1. Let p ∈ Gp \ p. Then, a necessary condition for VorGp(p) ′ and VorGp(p ) to have a common facet is: If p′ ∈ O1, that p′ lies in (the relative interior of) a facet of
2 VorO1 (p).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Proof of Delone’s bound (for Dirichlet stereohedra)
Let p be a base point for an orbit Gp, and consider separately its a translational orbits. That is, let Gp = O1 ∪ O2 ∪ ∪ Oa, with ′ p ∈ O1. Let p ∈ Gp \ p. Then, a necessary condition for VorGp(p) ′ and VorGp(p ) to have a common facet is: If p′ ∈ O1, that p′ lies in (the relative interior of) a facet of
2 VorO1 (p). ′ ′ If p ∈ O1, that p lies in the interior of 2VorO1 (p).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups intro. Proof of Delone’s bound (for Dirichlet stereohedra)
Let p be a base point for an orbit Gp, and consider separately its a translational orbits. That is, let Gp = O1 ∪ O2 ∪ ∪ Oa, with ′ p ∈ O1. Let p ∈ Gp \ p. Then, a necessary condition for VorGp(p) ′ and VorGp(p ) to have a common facet is: If p′ ∈ O1, that p′ lies in (the relative interior of) a facet of
2 VorO1 (p). ′ ′ If p ∈ O1, that p lies in the interior of 2VorO1 (p). A volume argument implies that the first happens for at most 2(2d − 1)times and the second at most 2d times. Adding up gives
a2d + 2d − 2
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Groups with reflections
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Reptiles have thick skins
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Reptiles have thick skins
The main reason why reflections help A LOT is that no reflection plane can cut a stereohedron.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Reptiles have thick skins
The main reason why reflections help A LOT is that no reflection plane can cut a stereohedron. Thus, each stereohedron is contained in a reflection cell (fundamental domain of the reflection subgroup of G) and may have two types of neighbors in VorGp(p):
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Reptiles have thick skins
The main reason why reflections help A LOT is that no reflection plane can cut a stereohedron. Thus, each stereohedron is contained in a reflection cell (fundamental domain of the reflection subgroup of G) and may have two types of neighbors in VorGp(p): Internal neighbors, lying in the same reflection cell as the base point p.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Reptiles have thick skins
The main reason why reflections help A LOT is that no reflection plane can cut a stereohedron. Thus, each stereohedron is contained in a reflection cell (fundamental domain of the reflection subgroup of G) and may have two types of neighbors in VorGp(p): Internal neighbors, lying in the same reflection cell as the base point p. External neighbors, sharing facets contained in reflection cells.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Reptiles have thick skins
The main reason why reflections help A LOT is that no reflection plane can cut a stereohedron. Thus, each stereohedron is contained in a reflection cell (fundamental domain of the reflection subgroup of G) and may have two types of neighbors in VorGp(p): Internal neighbors, lying in the same reflection cell as the base point p. External neighbors, sharing facets contained in reflection cells. There is at most one for each facet of the reflection cell.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Reptiles have thick skins
The main reason why reflections help A LOT is that no reflection plane can cut a stereohedron. Thus, each stereohedron is contained in a reflection cell (fundamental domain of the reflection subgroup of G) and may have two types of neighbors in VorGp(p): Internal neighbors, lying in the same reflection cell as the base point p. External neighbors, sharing facets contained in reflection cells. There is at most one for each facet of the reflection cell. We look separately at the cases of 1, 2, and 3 independent reflections (i.e., reflection cells in unbounded in two, one and zero dimensions respectively).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Let p ∈ R be our base point and reflection cell.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Let p ∈ R be our base point and reflection cell. R is a product of simplices: tetrahedron, triangular prism, or cube.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Let p ∈ R be our base point and reflection cell. R is a product of simplices: tetrahedron, triangular prism, or cube. Thus it produces at most 6 external neighbors.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Let p ∈ R be our base point and reflection cell. R is a product of simplices: tetrahedron, triangular prism, or cube. Thus it produces at most 6 external neighbors. All points of Gp ∩ R lie at the same distance from the centroid of R. Inside he reflection cell, Vor Gp(p) is (a cone over) the Voronoi diagram of an orbit of points in a 2-sphere.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Let p ∈ R be our base point and reflection cell. R is a product of simplices: tetrahedron, triangular prism, or cube. Thus it produces at most 6 external neighbors. All points of Gp ∩ R lie at the same distance from the centroid of R. Inside he reflection cell, Vor Gp(p) is (a cone over) the Voronoi diagram of an orbit of points in a 2-sphere.Thus, p has at most 5 internal neighbors.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Let p ∈ R be our base point and reflection cell. R is a product of simplices: tetrahedron, triangular prism, or cube. Thus it produces at most 6 external neighbors. All points of Gp ∩ R lie at the same distance from the centroid of R. Inside he reflection cell, Vor Gp(p) is (a cone over) the Voronoi diagram of an orbit of points in a 2-sphere.Thus, p has at most 5 internal neighbors.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Let p ∈ R be our base point and reflection cell. R is a product of simplices: tetrahedron, triangular prism, or cube. Thus it produces at most 6 external neighbors. All points of Gp ∩ R lie at the same distance from the centroid of R. Inside he reflection cell, Vor Gp(p) is (a cone over) the Voronoi diagram of an orbit of points in a 2-sphere.Thus, p has at most 5 internal neighbors. Both bounds are tight, but they interfere with one another:
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Let p ∈ R be our base point and reflection cell. R is a product of simplices: tetrahedron, triangular prism, or cube. Thus it produces at most 6 external neighbors. All points of Gp ∩ R lie at the same distance from the centroid of R. Inside he reflection cell, Vor Gp(p) is (a cone over) the Voronoi diagram of an orbit of points in a 2-sphere.Thus, p has at most 5 internal neighbors. Both bounds are tight, but they interfere with one another: if there are three or more internal neighbors, the stabilizer of the centroid of R must be one of:
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Let p ∈ R be our base point and reflection cell. R is a product of simplices: tetrahedron, triangular prism, or cube. Thus it produces at most 6 external neighbors. All points of Gp ∩ R lie at the same distance from the centroid of R. Inside he reflection cell, Vor Gp(p) is (a cone over) the Voronoi diagram of an orbit of points in a 2-sphere.Thus, p has at most 5 internal neighbors. Both bounds are tight, but they interfere with one another: if there are three or more internal neighbors, the stabilizer of the centroid of R must be one of: A dihedral group of order 4, 6 or 8.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Let p ∈ R be our base point and reflection cell. R is a product of simplices: tetrahedron, triangular prism, or cube. Thus it produces at most 6 external neighbors. All points of Gp ∩ R lie at the same distance from the centroid of R. Inside he reflection cell, Vor Gp(p) is (a cone over) the Voronoi diagram of an orbit of points in a 2-sphere.Thus, p has at most 5 internal neighbors. Both bounds are tight, but they interfere with one another: if there are three or more internal neighbors, the stabilizer of the centroid of R must be one of: A dihedral group of order 4, 6 or 8. This produces at most four internal neighbors and at most four external neighbors.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Let p ∈ R be our base point and reflection cell. R is a product of simplices: tetrahedron, triangular prism, or cube. Thus it produces at most 6 external neighbors. All points of Gp ∩ R lie at the same distance from the centroid of R. Inside he reflection cell, Vor Gp(p) is (a cone over) the Voronoi diagram of an orbit of points in a 2-sphere.Thus, p has at most 5 internal neighbors. Both bounds are tight, but they interfere with one another: if there are three or more internal neighbors, the stabilizer of the centroid of R must be one of: A dihedral group of order 4, 6 or 8. This produces at most four internal neighbors and at most four external neighbors. The group of orientation preserving symmetries of a regular tetrahedron or cube.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Let p ∈ R be our base point and reflection cell. R is a product of simplices: tetrahedron, triangular prism, or cube. Thus it produces at most 6 external neighbors. All points of Gp ∩ R lie at the same distance from the centroid of R. Inside he reflection cell, Vor Gp(p) is (a cone over) the Voronoi diagram of an orbit of points in a 2-sphere.Thus, p has at most 5 internal neighbors. Both bounds are tight, but they interfere with one another: if there are three or more internal neighbors, the stabilizer of the centroid of R must be one of: A dihedral group of order 4, 6 or 8. This produces at most four internal neighbors and at most four external neighbors. The group of orientation preserving symmetries of a regular tetrahedron or cube. At most three external neighbors.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Three independent reflections (reflection cell is bounded)
Summing up: Theorem (Bochis-S., 2001) Dirichlet stereohedra for groups with three independent reflections have at most eight facets. The bound is attained
7 v’ v’’ 3 54 1 7 3 3 1 6 8 2 7 2 82 8 1 v
4(a)2 2 (b) (c) P m m m F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 2-d of reflections (reflection cell = unbounded prism)
R is now an unbounded prism over a triangle or quadrilateral, and G has translations parallel to its recession line (call it l).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 2-d of reflections (reflection cell = unbounded prism)
R is now an unbounded prism over a triangle or quadrilateral, and G has translations parallel to its recession line (call it l). Thus, Gp ∩ R decomposes as a union of 1-dimensional crystallographic orbits on lines parallel to l.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 2-d of reflections (reflection cell = unbounded prism)
R is now an unbounded prism over a triangle or quadrilateral, and G has translations parallel to its recession line (call it l). Thus, Gp ∩ R decomposes as a union of 1-dimensional crystallographic orbits on lines parallel to l. The number of such lines is at most 8 (symmetries of the square) and p has at most two neighbors along each of them.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 2-d of reflections (reflection cell = unbounded prism)
R is now an unbounded prism over a triangle or quadrilateral, and G has translations parallel to its recession line (call it l). Thus, Gp ∩ R decomposes as a union of 1-dimensional crystallographic orbits on lines parallel to l. The number of such lines is at most 8 (symmetries of the square) and p has at most two neighbors along each of them. Thus, there are at most 16 internal neighbors.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 2-d of reflections (reflection cell = unbounded prism)
R is now an unbounded prism over a triangle or quadrilateral, and G has translations parallel to its recession line (call it l). Thus, Gp ∩ R decomposes as a union of 1-dimensional crystallographic orbits on lines parallel to l. The number of such lines is at most 8 (symmetries of the square) and p has at most two neighbors along each of them. Thus, there are at most 16 internal neighbors. There can be four external neighbors (facets of the infinite prism)
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 2-d of reflections (reflection cell = unbounded prism)
R is now an unbounded prism over a triangle or quadrilateral, and G has translations parallel to its recession line (call it l). Thus, Gp ∩ R decomposes as a union of 1-dimensional crystallographic orbits on lines parallel to l. The number of such lines is at most 8 (symmetries of the square) and p has at most two neighbors along each of them. Thus, there are at most 16 internal neighbors. There can be four external neighbors (facets of the infinite prism) but, as before, four or more internal neighbors imply only two external neighbors. Thus:
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 2-d of reflections (reflection cell = unbounded prism)
Theorem (Bochis-S., 2001) Dirichlet stereohedra for groups with two independent reflections have at most 18 facets. The bound is attained.
3/4 3/4
1/2 1/4 1/2 1/4
41 222 2 I g c d
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 1-d of reflections (reflection cell = R2 × I )
R is the region between two parallel planes (call them horizontal).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 1-d of reflections (reflection cell = R2 × I )
R is the region between two parallel planes (call them horizontal). 2-h 1 h P R -h -1 h-2 All points of Gp ∩ R are at the same distance to the middle plane
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 1-d of reflections (reflection cell = R2 × I )
R is the region between two parallel planes (call them horizontal). 2-h 1 h P R -h -1 h-2 All points of Gp ∩ R are at the same distance to the middle plane and the points at each height form a 2-dimensional crystallographic orbit.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 1-d of reflections (reflection cell = R2 × I )
R is the region between two parallel planes (call them horizontal). 2-h 1 h P R -h -1 h-2 All points of Gp ∩ R are at the same distance to the middle plane and the points at each height form a 2-dimensional crystallographic orbit. p has at most six internal neighbors at its same height (planar Dirichlet region)
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 1-d of reflections (reflection cell = R2 × I )
R is the region between two parallel planes (call them horizontal). 2-h 1 h P R -h -1 h-2 All points of Gp ∩ R are at the same distance to the middle plane and the points at each height form a 2-dimensional crystallographic orbit. p has at most six internal neighbors at its same height (planar Dirichlet region) and at most seven (*) at the other height
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 1-d of reflections (reflection cell = R2 × I )
R is the region between two parallel planes (call them horizontal). 2-h 1 h P R -h -1 h-2 All points of Gp ∩ R are at the same distance to the middle plane and the points at each height form a 2-dimensional crystallographic orbit. p has at most six internal neighbors at its same height (planar Dirichlet region) and at most seven (*) at the other height, making a total of at most 13 internal neighbors.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 1-d of reflections (reflection cell = R2 × I )
R is the region between two parallel planes (call them horizontal). 2-h 1 h P R -h -1 h-2 All points of Gp ∩ R are at the same distance to the middle plane and the points at each height form a 2-dimensional crystallographic orbit. p has at most six internal neighbors at its same height (planar Dirichlet region) and at most seven (*) at the other height, making a total of at most 13 internal neighbors. There are, of course, at most two external neighbors. Thus:
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections 1-d of reflections (reflection cell = R2 × I )
Theorem (Bochis-S., 2001) Dirichlet stereohedra for groups with all reflections parallel have at most 15 facets. The bound is attained.
2 2 2 P m c c
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Intersection of two (planar) Dirichlet tesselations
2-h In the case of one reflection we said 1 “P has at most seven neighbors h P R at the other height”. -h -1 h-2
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Intersection of two (planar) Dirichlet tesselations
2-h In the case of one reflection we said 1 “P has at most seven neighbors h P R at the other height”. -h -1 This follows from: h-2
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Intersection of two (planar) Dirichlet tesselations
2-h In the case of one reflection we said 1 “P has at most seven neighbors h P R at the other height”. -h -1 This follows from: h-2 ′ VorGP (P) and VorGP (P ) with P ∈ {z = h} and ′ ′ P ∈ {z = h } can share a facet only if VorGP∩{z=h}(P) and ′ VorGP∩{z=h′}(P ) overlap.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Intersection of two (planar) Dirichlet tesselations
2-h In the case of one reflection we said 1 “P has at most seven neighbors h P R at the other height”. -h -1 This follows from: h-2 ′ VorGP (P) and VorGP (P ) with P ∈ {z = h} and ′ ′ P ∈ {z = h } can share a facet only if VorGP∩{z=h}(P) and ′ VorGP∩{z=h′}(P ) overlap.
VorGP∩{z=h} and VorGP∩{z=h′} are (infinite prisms over) planar Dirichlet tesselations for a certain group Gh (the group go horizontal motions in G).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Groups with reflections Intersection of two (planar) Dirichlet tesselations
2-h In the case of one reflection we said 1 “P has at most seven neighbors h P R at the other height”. -h -1 This follows from: h-2 ′ VorGP (P) and VorGP (P ) with P ∈ {z = h} and ′ ′ P ∈ {z = h } can share a facet only if VorGP∩{z=h}(P) and ′ VorGP∩{z=h′}(P ) overlap.
VorGP∩{z=h} and VorGP∩{z=h′} are (infinite prisms over) planar Dirichlet tesselations for a certain group Gh (the group go horizontal motions in G).
A planar Dirichlet region VorGhp(GhP) can intersect at most ′ seven regions of a Dirichlet tessellation VorGh (Ghp ) for the same crystallographic group. (Non-trivial!!!)
F. Santos Number of facets of 3-d Dirichlet stereohedra
Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups Fish and birds come in layers
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups Fish and birds come in layers
Non-cubic groups all have one (or more) special direction that is not mixed with the other two.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups Fish and birds come in layers
Non-cubic groups all have one (or more) special direction that is not mixed with the other two. (That is: every non-cubic group G is a subgroup of a Cartesian product G1 × G2, where Gi is a crystallographic group of dimension i). We think of G2 as horizontal and G1 as vertical.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups Fish and birds come in layers
Non-cubic groups all have one (or more) special direction that is not mixed with the other two. (That is: every non-cubic group G is a subgroup of a Cartesian product G1 × G2, where Gi is a crystallographic group of dimension i). We think of G2 as horizontal and G1 as vertical. If “many” aspects are at the same height (that is, if the horizontal subgroup G0 := G ∩ (1 × G2) has many aspects) we can take advantage of the fact that “P has at most seven neighbors at each other height”.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups Fish and birds come in layers
Non-cubic groups all have one (or more) special direction that is not mixed with the other two. (That is: every non-cubic group G is a subgroup of a Cartesian product G1 × G2, where Gi is a crystallographic group of dimension i). We think of G2 as horizontal and G1 as vertical. If “many” aspects are at the same height (that is, if the horizontal subgroup G0 := G ∩ (1 × G2) has many aspects) we can take advantage of the fact that “P has at most seven neighbors at each other height”. And, the “seven” is an upper bound.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups Fish and birds come in layers
Non-cubic groups all have one (or more) special direction that is not mixed with the other two. (That is: every non-cubic group G is a subgroup of a Cartesian product G1 × G2, where Gi is a crystallographic group of dimension i). We think of G2 as horizontal and G1 as vertical. If “many” aspects are at the same height (that is, if the horizontal subgroup G0 := G ∩ (1 × G2) has many aspects) we can take advantage of the fact that “P has at most seven neighbors at each other height”. And, the “seven” is an upper bound. Depending on the type of the horizontal group G0 we can use:
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups Fish and birds come in layers
Non-cubic groups all have one (or more) special direction that is not mixed with the other two. (That is: every non-cubic group G is a subgroup of a Cartesian product G1 × G2, where Gi is a crystallographic group of dimension i). We think of G2 as horizontal and G1 as vertical. If “many” aspects are at the same height (that is, if the horizontal subgroup G0 := G ∩ (1 × G2) has many aspects) we can take advantage of the fact that “P has at most seven neighbors at each other height”. And, the “seven” is an upper bound. Depending on the type of the horizontal group G0 we can use: 4 for p1, p3, p4 and p6, 7 for p2, pg and pgg.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
1/2 Non-cubic groups 1/2 1/2 1/2 1/2 1/6 4/6 Some examples 1/2 4/6 1/6 1/6 4/6 1/6 1/2 1/2 4/6 5/6 4/6 1/2 1/2 3/6 5/6 1/6 1/6 3/6 1/2 1/2 4/6 1/2 1/2 5/6 3/6 3/6 5/6 1/2 1/2 3/6 1/2 1/2 5/6 3/6 5/6
Hexagonal groups with 12 aspects and no reflections
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
1/2 Non-cubic groups 1/2 1/2 1/2 1/2 1/6 4/6 Some examples 1/2 4/6 1/6 1/6 4/6 1/6 1/2 1/2 4/6 5/6 4/6 1/2 1/2 3/6 5/6 1/6 1/6 3/6 1/2 1/2 4/6 1/2 1/2 5/6 3/6 3/6 5/6 1/2 1/2 3/6 1/2 1/2 5/6 3/6 5/6
Hexagonal groups with 12 aspects and no reflections In the first two, there are three horizontal planes (×2, we have to count one above and one below p) other than the one of the base point.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
1/2 Non-cubic groups 1/2 1/2 1/2 1/2 1/6 4/6 Some examples 1/2 4/6 1/6 1/6 4/6 1/6 1/2 1/2 4/6 5/6 4/6 1/2 1/2 3/6 5/6 1/6 1/6 3/6 1/2 1/2 4/6 1/2 1/2 5/6 3/6 3/6 5/6 1/2 1/2 3/6 1/2 1/2 5/6 3/6 5/6
Hexagonal groups with 12 aspects and no reflections In the first two, there are three horizontal planes (×2, we have to count one above and one below p) other than the one of the base point. Since G0 is a p3, we count four neighbours in each of them (24 so far)
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
1/2 Non-cubic groups 1/2 1/2 1/2 1/2 1/6 4/6 Some examples 1/2 4/6 1/6 1/6 4/6 1/6 1/2 1/2 4/6 5/6 4/6 1/2 1/2 3/6 5/6 1/6 1/6 3/6 1/2 1/2 4/6 1/2 1/2 5/6 3/6 3/6 5/6 1/2 1/2 3/6 1/2 1/2 5/6 3/6 5/6
Hexagonal groups with 12 aspects and no reflections In the first two, there are three horizontal planes (×2, we have to count one above and one below p) other than the one of the base point. Since G0 is a p3, we count four neighbours in each of them (24 so far) plus six on the base of p,
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
1/2 Non-cubic groups 1/2 1/2 1/2 1/2 1/6 4/6 Some examples 1/2 4/6 1/6 1/6 4/6 1/6 1/2 1/2 4/6 5/6 4/6 1/2 1/2 3/6 5/6 1/6 1/6 3/6 1/2 1/2 4/6 1/2 1/2 5/6 3/6 3/6 5/6 1/2 1/2 3/6 1/2 1/2 5/6 3/6 5/6
Hexagonal groups with 12 aspects and no reflections In the first two, there are three horizontal planes (×2, we have to count one above and one below p) other than the one of the base point. Since G0 is a p3, we count four neighbours in each of them (24 so far) plus six on the base of p, plus the two points directly above and below p.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
1/2 Non-cubic groups 1/2 1/2 1/2 1/2 1/6 4/6 Some examples 1/2 4/6 1/6 1/6 4/6 1/6 1/2 1/2 4/6 5/6 4/6 1/2 1/2 3/6 5/6 1/6 1/6 3/6 1/2 1/2 4/6 1/2 1/2 5/6 3/6 3/6 5/6 1/2 1/2 3/6 1/2 1/2 5/6 3/6 5/6
Hexagonal groups with 12 aspects and no reflections In the first two, there are three horizontal planes (×2, we have to count one above and one below p) other than the one of the base point. Since G0 is a p3, we count four neighbours in each of them (24 so far) plus six on the base of p, plus the two points directly above and below p. Total: at most 32 neighbors.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
1/2 Non-cubic groups 1/2 1/2 1/2 1/2 1/6 4/6 Some examples 1/2 4/6 1/6 1/6 4/6 1/6 1/2 1/2 4/6 5/6 4/6 1/2 1/2 3/6 5/6 1/6 1/6 3/6 1/2 1/2 4/6 1/2 1/2 5/6 3/6 3/6 5/6 1/2 1/2 3/6 1/2 1/2 5/6 3/6 5/6
Hexagonal groups with 12 aspects and no reflections In the first two, there are three horizontal planes (×2, we have to count one above and one below p) other than the one of the base point. Since G0 is a p3, we count four neighbours in each of them (24 so far) plus six on the base of p, plus the two points directly above and below p. Total: at most 32 neighbors. In the last one we have 11 × 2 instead of 3 × 2 horizontal planes, giving a bound of 96.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
1/2 Non-cubic groups 1/2 1/2 1/2 1/2 1/6 4/6 Some examples 1/2 4/6 1/6 1/6 4/6 1/6 1/2 1/2 4/6 5/6 4/6 1/2 1/2 3/6 5/6 1/6 1/6 3/6 1/2 1/2 4/6 1/2 1/2 5/6 3/6 3/6 5/6 1/2 1/2 3/6 1/2 1/2 5/6 3/6 5/6
Hexagonal groups with 12 aspects and no reflections In the first two, there are three horizontal planes (×2, we have to count one above and one below p) other than the one of the base point. Since G0 is a p3, we count four neighbours in each of them (24 so far) plus six on the base of p, plus the two points directly above and below p. Total: at most 32 neighbors. In the last one we have 11 × 2 instead of 3 × 2 horizontal planes, giving a bound of 96. The extra ones come from the lattice not being primitive, but rhombohedral.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
1/2 Non-cubic groups 1/2 1/2 1/2 1/2 1/6 4/6 Some examples 1/2 4/6 1/6 1/6 4/6 1/6 1/2 1/2 4/6 5/6 4/6 1/2 1/2 3/6 5/6 1/6 1/6 3/6 1/2 1/2 4/6 1/2 1/2 5/6 3/6 3/6 5/6 1/2 1/2 3/6 1/2 1/2 5/6 3/6 5/6
Hexagonal groups with 12 aspects and no reflections In the first two, there are three horizontal planes (×2, we have to count one above and one below p) other than the one of the base point. Since G0 is a p3, we count four neighbours in each of them (24 so far) plus six on the base of p, plus the two points directly above and below p. Total: at most 32 neighbors. In the last one we have 11 × 2 instead of 3 × 2 horizontal planes, giving a bound of 96. The extra ones come from the lattice not being primitive, but rhombohedral. Each aspect of G splits into six instead of two aspects of G0
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first bound, for each group
For each non-cubic group G without reflections, let G0 be its horizontal subgroup, and consider the following parameters:
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first bound, for each group
For each non-cubic group G without reflections, let G0 be its horizontal subgroup, and consider the following parameters: a and a0, the numbers of aspects of G and G0.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first bound, for each group
For each non-cubic group G without reflections, let G0 be its horizontal subgroup, and consider the following parameters: a and a0, the numbers of aspects of G and G0. a parameter l depending on the lattice: 2 for primitive or “base centered”, 4 for face centered or body centered, 6 for rhombohedral.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first bound, for each group
For each non-cubic group G without reflections, let G0 be its horizontal subgroup, and consider the following parameters: a and a0, the numbers of aspects of G and G0. a parameter l depending on the lattice: 2 for primitive or “base centered”, 4 for face centered or body centered, 6 for rhombohedral. a parameter i depending on the type of G0: 4 for p1, p3, p4 and p6, and 7 for p2, pg and pgg.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first bound, for each group
For each non-cubic group G without reflections, let G0 be its horizontal subgroup, and consider the following parameters: a and a0, the numbers of aspects of G and G0. a parameter l depending on the lattice: 2 for primitive or “base centered”, 4 for face centered or body centered, 6 for rhombohedral. a parameter i depending on the type of G0: 4 for p1, p3, p4 and p6, and 7 for p2, pg and pgg.
Corollary A stereohedron for G cannot have more than i a l − 1 + 8 a0 neighbors.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first upper bound, for each group
With this we have that, of the 97 non-cubic groups without reflections:
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first upper bound, for each group
With this we have that, of the 97 non-cubic groups without reflections: 39 of them have four or less aspects. Delone bound is already ≤ 38, so we do not care much about these.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first upper bound, for each group
With this we have that, of the 97 non-cubic groups without reflections: 39 of them have four or less aspects. Delone bound is already ≤ 38, so we do not care much about these. In 24 more the corollary gives a bound ≤ 38. (Only 34 groups left).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first upper bound, for each group
With this we have that, of the 97 non-cubic groups without reflections: 39 of them have four or less aspects. Delone bound is already ≤ 38, so we do not care much about these. In 24 more the corollary gives a bound ≤ 38. (Only 34 groups left). The bound of the corollary is > 50 in only ten of the groups.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first upper bound, for each group
With this we have that, of the 97 non-cubic groups without reflections: 39 of them have four or less aspects. Delone bound is already ≤ 38, so we do not care much about these. In 24 more the corollary gives a bound ≤ 38. (Only 34 groups left). The bound of the corollary is > 50 in only ten of the groups. The worst bound is 106, obtained in the tetragonal group 41 2 2 I g c d .
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first upper bound, for each group
With this we have that, of the 97 non-cubic groups without reflections: 39 of them have four or less aspects. Delone bound is already ≤ 38, so we do not care much about these. In 24 more the corollary gives a bound ≤ 38. (Only 34 groups left). The bound of the corollary is > 50 in only ten of the groups. The worst bound is 106, obtained in the tetragonal group 41 2 2 I g c d . The second worst is 96, in the hexagonal group P6122 2 and the tetragonal group R3 c .
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first upper bound, for each group
With this we have that, of the 97 non-cubic groups without reflections: 39 of them have four or less aspects. Delone bound is already ≤ 38, so we do not care much about these. In 24 more the corollary gives a bound ≤ 38. (Only 34 groups left). The bound of the corollary is > 50 in only ten of the groups. The worst bound is 106, obtained in the tetragonal group 41 2 2 I g c d . The second worst is 96, in the hexagonal group P6122 2 and the tetragonal group R3 c . In the light of this, we concentrate in reducing the global upper bound of 106, and in trying to get more groups have bounds ≤ 38.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups A first upper bound, for each group
With this we have that, of the 97 non-cubic groups without reflections: 39 of them have four or less aspects. Delone bound is already ≤ 38, so we do not care much about these. In 24 more the corollary gives a bound ≤ 38. (Only 34 groups left). The bound of the corollary is > 50 in only ten of the groups. The worst bound is 106, obtained in the tetragonal group 41 2 2 I g c d . The second worst is 96, in the hexagonal group P6122 2 and the tetragonal group R3 c . In the light of this, we concentrate in reducing the global upper bound of 106, and in trying to get more groups have bounds ≤ 38. After some hand-waving:
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups The final upper bounds
Theorem (Bochis-S., 2006) Only 21 non-cubic groups can perhaps produce Dirichlet stereohedra with more than 38 facets. Only the following 9 can produce them with more than 50 facets:
Group Aspects Planar group Upper bound P41212 8 p1 64 41 I g 8 p2 70 I 4122 8 p2 70 I 42d 8 p2 70 2 2 2 F d d d 8 p2 70 P6222 12 p2 78 P6122 12 p1 78 2 R3 c 12 p3 79 41 2 2 I g c d 16 pgg 80
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups How good are the bounds?
In the example for groups with reflections in two dimensions we used: Lemma Let S be a set of points placed on a helix curve in R3:
c(t)=(r cos t, r sin t, ht).
Then, every two points c(t1), c(t2) ∈ S with |t1 − t2|≤ 2π form an edge in Del(S).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups How good are the bounds?
In the example for groups with reflections in two dimensions we used: Lemma Let S be a set of points placed on a helix curve in R3:
c(t)=(r cos t, r sin t, ht).
Then, every two points c(t1), c(t2) ∈ S with |t1 − t2|≤ 2π form an edge in Del(S).
This means that the appearance of screw-rotations in a crystallographic group is not only “bad for our methods”
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups How good are the bounds?
In the example for groups with reflections in two dimensions we used: Lemma Let S be a set of points placed on a helix curve in R3:
c(t)=(r cos t, r sin t, ht).
Then, every two points c(t1), c(t2) ∈ S with |t1 − t2|≤ 2π form an edge in Del(S).
This means that the appearance of screw-rotations in a crystallographic group is not only “bad for our methods” (they produce “sparse horizontal planes”).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups How good are the bounds?
In the example for groups with reflections in two dimensions we used: Lemma Let S be a set of points placed on a helix curve in R3:
c(t)=(r cos t, r sin t, ht).
Then, every two points c(t1), c(t2) ∈ S with |t1 − t2|≤ 2π form an edge in Del(S).
This means that the appearance of screw-rotations in a crystallographic group is not only “bad for our methods” (they produce “sparse horizontal planes”). It does favor existence of Dirichlet stereohedra with “many neighbors”.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Non-cubic groups How good are the bounds?
Theorem (Bochis-S. 2006)
41 There are Dirichlet stereohedra for the groups I 2 2 and P6122 with 28 and 32 facets, respectively.
Y
4/6 3/6 4 3/6 1/2 4/6 3/4 3/4 2/6 3 5/6 3/4 3/4 2/6 1/2 5/6 2 1/2 1/6 1/4 1/4 1 1/4 1/6 1/4 1/2
1 2 3 4 X I 4122 P6122
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Cubic groups
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Pets or beasts?
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Pets or beasts?
There are 35 cubic-groups, 22 of them without reflections.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Pets or beasts?
They do not have a “special direction” and, since they have ternary rotations in the diagonals of the cube cell, they have too many horizontal planes (up to 24) containing orbit points for the previous method to be of (much) help.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Pets or beasts?
They do not have a “special direction” and, since they have ternary rotations in the diagonals of the cube cell, they have too many horizontal planes (up to 24) containing orbit points for the previous method to be of (much) help. Somehow surprisingly, Conway, Delgado Friedrichs, Huson, and Thurston (2001) found a way to put “order in chaos”.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Pets or beasts?
They do not have a “special direction” and, since they have ternary rotations in the diagonals of the cube cell, they have too many horizontal planes (up to 24) containing orbit points for the previous method to be of (much) help. Somehow surprisingly, Conway, Delgado Friedrichs, Huson, and Thurston (2001) found a way to put “order in chaos”. Definition The odd subgroup of a cubic group is the one generated by its rotations of order three. Trivial fact: the odd subgroup has 12 aspects. In fact, its point group is 23 (orientation preserving symmetries of the tetrahedron)
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups The new classification of cubic groups
Theorem (Conway et al. 2001) Let G be a cubic group, and let O be its odd subgroup. Then: O is either the group F 23 or the group P213. G lies between O and its normalizer N (O).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups The new classification of cubic groups
Theorem (Conway et al. 2001) Let G be a cubic group, and let O be its odd subgroup. Then: O is either the group F 23 or the group P213. G lies between O and its normalizer N (O).
Definition A cubic group is called full or quarter depending on whether its odd subgroup is F 23 or P213.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups The new classification of cubic groups
Theorem (Conway et al. 2001) Let G be a cubic group, and let O be its odd subgroup. Then: O is either the group F 23 or the group P213. G lies between O and its normalizer N (O).
Definition A cubic group is called full or quarter depending on whether its odd subgroup is F 23 or P213.
Corollary The classification of full (resp. quarter) groups equals the classification of subgroups of the quotient N (F 23)/F (23) (resp. N (P213)/P213).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups The odd groups
The odd group F 23 is generated by the ternary rotations in all diagonals of a cubic grid.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups The odd groups
The odd group F 23 is generated by the ternary rotations in all diagonals of a cubic grid. P213, in contrast, contains only one fourth (a quarter) of the ternary rotation axes
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups The odd groups
The odd group F 23 is generated by the ternary rotations in all diagonals of a cubic grid. P213, in contrast, contains only one fourth (a quarter) of the ternary rotation axes (one through each vertex in the grid).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups The odd groups
The odd group F 23 is generated by the ternary rotations in all diagonals of a cubic grid. P213, in contrast, contains only one fourth (a quarter) of the ternary rotation axes (one through each vertex in the grid).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Pets and beasts
Full groups have four ternary rotations intersecting at each vertex of the grid. That is good; it implies some big Delaunay cells which “block” many possible neighbors.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Pets and beasts
Full groups have four ternary rotations intersecting at each vertex of the grid. That is good; it implies some big Delaunay cells which “block” many possible neighbors. Also, full groups often have rotations of order four (and 13 of them contain reflections).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Pets and beasts
Full groups have four ternary rotations intersecting at each vertex of the grid. That is good; it implies some big Delaunay cells which “block” many possible neighbors. Also, full groups often have rotations of order four (and 13 of them contain reflections).
Quarter groups never contain rotations of order four (or reflections).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Pets and beasts
Full groups have four ternary rotations intersecting at each vertex of the grid. That is good; it implies some big Delaunay cells which “block” many possible neighbors. Also, full groups often have rotations of order four (and 13 of them contain reflections).
Quarter groups never contain rotations of order four (or reflections). Instead, they contain screw rotations of orders two or four, which is bad.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Pets and beasts
Full groups have four ternary rotations intersecting at each vertex of the grid. That is good; it implies some big Delaunay cells which “block” many possible neighbors. Also, full groups often have rotations of order four (and 13 of them contain reflections).
Quarter groups never contain rotations of order four (or reflections). Instead, they contain screw rotations of orders two or four, which is bad. No point has a stabilizer of order greater than three, so Delaunay cells tend to be small.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Understanding full groups
To understand full groups, the Delaunay triangulation of the body-centered cubic lattice is useful.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Understanding full groups
To understand full groups, the Delaunay triangulation of the body-centered cubic lattice is useful. Its cells are tetrahedra with two opposite edges in coordinate directions and the other three in diagonals of the cubic grid.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Understanding full groups
To understand full groups, the Delaunay triangulation of the body-centered cubic lattice is useful. Its cells are tetrahedra with two opposite edges in coordinate directions and the other three in diagonals of the cubic grid. Dihedral angles at these edges are π/2 and π/3 respectively.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Understanding full groups
To understand full groups, the Delaunay triangulation of the body-centered cubic lattice is useful. Its cells are tetrahedra with two opposite edges in coordinate directions and the other three in diagonals of the cubic grid. Dihedral angles at these edges are π/2 and π/3 respectively. It is a balanced triangulation (Points can be labeled 1 to 4 with every tetrahedron getting a vertex of each label. Tetrahedra can be colored black and white so that adjacent ones have opposite color).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Understanding full groups
To understand full groups, the Delaunay triangulation of the body-centered cubic lattice is useful. Its cells are tetrahedra with two opposite edges in coordinate directions and the other three in diagonals of the cubic grid. Dihedral angles at these edges are π/2 and π/3 respectively. It is a balanced triangulation (Points can be labeled 1 to 4 with every tetrahedron getting a vertex of each label. Tetrahedra can be colored black and white so that adjacent ones have opposite color). The odd group F 23 is the group of orientation-preserving automorphisms of the labelled triangulation.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Understanding full groups
To understand full groups, the Delaunay triangulation of the body-centered cubic lattice is useful. Its cells are tetrahedra with two opposite edges in coordinate directions and the other three in diagonals of the cubic grid. Dihedral angles at these edges are π/2 and π/3 respectively. It is a balanced triangulation (Points can be labeled 1 to 4 with every tetrahedron getting a vertex of each label. Tetrahedra can be colored black and white so that adjacent ones have opposite color). The odd group F 23 is the group of orientation-preserving automorphisms of the labelled triangulation. Equivalently, the group of labelled automorphisms that preserve color of tetrahedra.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Understanding full groups
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Understanding full groups
Every pair of adjacent tetrahedra form a fundamental domain of F 23.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Understanding full groups
Every pair of adjacent tetrahedra form a fundamental domain of F 23. A fundamental domain of N (F 23) is obtained slicing each tetrahedron in eight parts by four planes containing the mid points of the two “long” edges. This suggests the following strategy to bound the number of facets of Dirichlet stereohedra:
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups A first upper bound
Suppose that your base point p lies in a certain (white) tetrahedron T0.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups A first upper bound
Suppose that your base point p lies in a certain (white) tetrahedron T0. Since G contains rotations of order three (resp. two) at the short (resp. long) edges of T0, VorGp(p) is contained in the union of T0 and its four adjacent (black) tetrahedra, T1, ..., T4.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups A first upper bound
Suppose that your base point p lies in a certain (white) tetrahedron T0. Since G contains rotations of order three (resp. two) at the short (resp. long) edges of T0, VorGp(p) is contained in the union of T0 and its four adjacent (black) tetrahedra, T1, ..., T4. We call this union the extended Voronoi region of T0, and denote it ExtVorG (T0).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups A first upper bound
Suppose that your base point p lies in a certain (white) tetrahedron T0. Since G contains rotations of order three (resp. two) at the short (resp. long) edges of T0, VorGp(p) is contained in the union of T0 and its four adjacent (black) tetrahedra, T1, ..., T4. We call this union the extended Voronoi region of T0, and denote it ExtVorG (T0). Since that holds for every p′ ∈ Gp, for p and p′ to be neighbors it is necessary that the extended Voronoi regions of T and the tetrahedron T ′ containing p′ overlap.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups A first upper bound
Suppose that your base point p lies in a certain (white) tetrahedron T0. Since G contains rotations of order three (resp. two) at the short (resp. long) edges of T0, VorGp(p) is contained in the union of T0 and its four adjacent (black) tetrahedra, T1, ..., T4. We call this union the extended Voronoi region of T0, and denote it ExtVorG (T0). Since that holds for every p′ ∈ Gp, for p and p′ to be neighbors it is necessary that the extended Voronoi regions of T and the tetrahedron T ′ containing p′ overlap. That is, T ′ is one of T0,..., T4 or the other 10 white tetrahedra adjacent to them.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups A first upper bound
Suppose that your base point p lies in a certain (white) tetrahedron T0. Since G contains rotations of order three (resp. two) at the short (resp. long) edges of T0, VorGp(p) is contained in the union of T0 and its four adjacent (black) tetrahedra, T1, ..., T4. We call this union the extended Voronoi region of T0, and denote it ExtVorG (T0). Since that holds for every p′ ∈ Gp, for p and p′ to be neighbors it is necessary that the extended Voronoi regions of T and the tetrahedron T ′ containing p′ overlap. That is, T ′ is one of T0,..., T4 or the other 10 white tetrahedra adjacent to them.We call this union the influence region of T0.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups A first upper bound
The influence region has 4 black and 11 white tetrahedra that can possibly contain neighbors of p.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups A first upper bound
The influence region has 4 black and 11 white tetrahedra that can possibly contain neighbors of p. This gives an upper bound of 10 facets for the Dirichlet stereohedra of T0 and an upper bound of 14 for the other four full groups with one orbit point per tetrahedron
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups A first upper bound
The influence region has 4 black and 11 white tetrahedra that can possibly contain neighbors of p. This gives an upper bound of 10 facets for the Dirichlet stereohedra of T0 and an upper bound of 14 for the other four full groups with one orbit point per tetrahedron, and a global upper bound of 59 neighbors for full groups without reflections.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups A first upper bound
The influence region has 4 black and 11 white tetrahedra that can possibly contain neighbors of p. This gives an upper bound of 10 facets for the Dirichlet stereohedra of T0 and an upper bound of 14 for the other four full groups with one orbit point per tetrahedron, and a global upper bound of 59 neighbors for full groups without reflections. More precisely: Lemma The number of facets of a full-cubic group G wo. reflections cannot exceed (11 + 4m)s − 1, where s ∈ {1, 2, 4} is the number of orbit points per white tetrahedron and m ∈ {0, 1} indicates whether G mixes colors of tetrahedra or not.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups The 27 full groups
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups The 27 full groups
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Refined bound
The worst value of this bound is 59, achieved in two groups I 432 4 2 and P n 3 n that contain rotations of order four on the two long edges of T0.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Refined bound
The worst value of this bound is 59, achieved in two groups I 432 4 2 and P n 3 n that contain rotations of order four on the two long edges of T0. These rotations can be used to make the extended Voronoi region and influence region smaller.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Refined bound
The worst value of this bound is 59, achieved in two groups I 432 4 2 and P n 3 n that contain rotations of order four on the two long edges of T0. These rotations can be used to make the extended Voronoi region and influence region smaller. Consider the base tetrahedron T0 divided into eight smaller tetrahedra A,...,H by four planes in the natural (symmetric) way.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Refined bound
The worst value of this bound is 59, achieved in two groups I 432 4 2 and P n 3 n that contain rotations of order four on the two long edges of T0. These rotations can be used to make the extended Voronoi region and influence region smaller. Consider the base tetrahedron T0 divided into eight smaller tetrahedra A,...,H by four planes in the natural (symmetric) way. These eight smaller tetrahedra (call them fundamental subdomains) are fundamental domains of the normalizer N (F 23) ≥ G.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Refined bound
The worst value of this bound is 59, achieved in two groups I 432 4 2 and P n 3 n that contain rotations of order four on the two long edges of T0. These rotations can be used to make the extended Voronoi region and influence region smaller. Consider the base tetrahedron T0 divided into eight smaller tetrahedra A,...,H by four planes in the natural (symmetric) way. These eight smaller tetrahedra (call them fundamental subdomains) are fundamental domains of the normalizer N (F 23) ≥ G. Wlog assume p ∈ A.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Refined bound
The worst value of this bound is 59, achieved in two groups I 432 4 2 and P n 3 n that contain rotations of order four on the two long edges of T0. These rotations can be used to make the extended Voronoi region and influence region smaller. Consider the base tetrahedron T0 divided into eight smaller tetrahedra A,...,H by four planes in the natural (symmetric) way. These eight smaller tetrahedra (call them fundamental subdomains) are fundamental domains of the normalizer N (F 23) ≥ G. Wlog assume p ∈ A. Then, assuming G contains the order four rotations, the extended Voronoi region of A consists of only 13 (instead of the former 8 × 5 = 40 fundamental subdomains.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Refined bound
The worst value of this bound is 59, achieved in two groups I 432 4 2 and P n 3 n that contain rotations of order four on the two long edges of T0. These rotations can be used to make the extended Voronoi region and influence region smaller. Consider the base tetrahedron T0 divided into eight smaller tetrahedra A,...,H by four planes in the natural (symmetric) way. These eight smaller tetrahedra (call them fundamental subdomains) are fundamental domains of the normalizer N (F 23) ≥ G. Wlog assume p ∈ A. Then, assuming G contains the order four rotations, the extended Voronoi region of A consists of only 13 (instead of the former 8 × 5 = 40 fundamental subdomains. The influence region of A consists of only 44 fundamental subdomains (instead of the former 8 × 15 = 120.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups
Refined bound v’3
D E
C F v2 v4 B G
A H T3
v1 v1 v1 v1
AH AH H A
G B B G G B v2 v4 v’4 v v v’2 F C 2 C F 4 F C
ED D E DE T4 T0 T2
v3 v3 v3 v3
D E
C F v2 v4 B G
A H T1
v’1
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
v’’31
Cubic groups = T31 T13 HA
v’3 B G v’3 v2 v4 C F Refined bound E D E D D E F C F C v’’34 v2 v4 v’’32 G B v’3 G B H A v’3 H A T34 T32 v’’43 v’’23 D E v1 v1 DE C F DE v2 v4 F C B G F C v’4 v2 v4 v’2 G B A H G B T3 H A H A T43 v1 T23 v v1 1 v1 v1 v1 v1 v1
AH AH AH H A AH
B G G B B G G B B G v’4 v2 v4 v’2 v’’42 v’4 v2 v4 v’2 v’’24 C F F C C F F C C F
ED D E DE = DE DE = T42 T24 T4 T0 T2 T24 T42 v3 v v3 v3 v3 3 v3 v3 v3 D E D DE
F C D E F C v’4 v2 v4 v’2 G B C F G B v2 v4 H A B G AH T41 T21 v3 A H v3 T1 v’’41 v’’21 D E D v’1 E D F C v’1 F C v’’14 v2 v4 v’’12 G B G B AH H A AH T14 B G T12 v2 v4 CF v’1 v’1 D E = T13 T31
v’’13
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Cubic groups Refined bound
Theorem (Sabariego-S. 2009) Dirichlet stereohedra for the 22 full-groups without reflections cannot have more than 23 facets. More precisely:
Group (m, s) Bound Group (m, s) Bound P432 (1, 2) 11 F 23 (0, 1) 10 I 23 (1, 2) 21 2 F 432 (1, 1) 14 P n 3 (1, 2) 23 41 2 F 43c (1, 1) 14 F d 3 n (1, 2) 25 2 P43n (0, 4) 23 F d 3 (1, 1) 14 P23 (0, 2) 15 P4232 (0, 4) 25 4 2 1 F 4 32 (0, 2) 17 P n 3 n (1, 4) 23 I 432 (1, 4) 22
F. Santos Number of facets of 3-d Dirichlet stereohedra
Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups The 8 quarter groups X X 1/2 1/4 3/4 1/2 1/2 1/2 1/2 3/4 1/4 32 3/4 32 1/2 1 1 4 4 I P 1/4 3/4 1/4 1/43/4 1/4 3/4 Y Y X X X X 2 d 1/2 1/2 1/2 3/4 1/4 3/4 1/4 3 1/2 1/2 1/2 1/2 3 1 g 1 4 2 d I 1/2 1/2 1/2 1/2 3 3/4 ′ 1/2 1/2 1/2 3/43/4 P 2 43 I I = )= 3/4 1/4 1/4 Q 1/2 1/2 Q ( N 1/2 1/2 3/4 1/4 3/4 1/4 1/4 1/4 3/4 3/43/4 1/4 1/41/4 1/4 3/4 Y Y Y Y ւ↓ց ց↓ւ X X 1/2 1/2 3 1/2 3 1/2 1 a 2 g 2 ↓ց↓ւ↓ I P 1/2 1/2 1/2 1/2 1/21/2 1/2 1/2 Y Y
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Outline of the method
The general idea is to adapt the “extended Voronoi region / influence region” method:
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Outline of the method
The general idea is to adapt the “extended Voronoi region / influence region” method:
1 Assume that your base point lies in a certain domain T0, that we call a “fundamental subdomain”.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Outline of the method
The general idea is to adapt the “extended Voronoi region / influence region” method:
1 Assume that your base point lies in a certain domain T0, that we call a “fundamental subdomain”. If D is (or contains) a fundamental domain F of N (G) this is no loss of generality. But you can also cover N (G) by several fundamental subdomains D1,..., Dn and repeat the process below for each (pair of) Di .
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Outline of the method
The general idea is to adapt the “extended Voronoi region / influence region” method:
1 Assume that your base point lies in a certain domain T0, that we call a “fundamental subdomain”. 2 Compute an extended Voronoi region of D, ExtVorG (D); this is any region guaranteed to contain VorGp(p) for every p ∈ D.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Outline of the method
The general idea is to adapt the “extended Voronoi region / influence region” method:
1 Assume that your base point lies in a certain domain T0, that we call a “fundamental subdomain”. 2 Compute an extended Voronoi region of D, ExtVorG (D); this is any region guaranteed to contain VorGp(p) for every p ∈ D.
To bound ExtVorG (D) use motions that are in your group (preferably translations and rotations). Remark: ExtVorG (D) may not be convex (specially if you used rotations of order 2 to cut it out).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Outline of the method
The general idea is to adapt the “extended Voronoi region / influence region” method:
1 Assume that your base point lies in a certain domain T0, that we call a “fundamental subdomain”. 2 Compute an extended Voronoi region of D, ExtVorG (D); this is any region guaranteed to contain VorGp(p) for every p ∈ D. 3 Find all other fundamental subdomains D′ (of all types, if D does not contain a fundamental domain of N (G)) such that ExtVorG (D) ∩ ′ ExtVorG (D ) is not empty. Call InflG (D) the union of them.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Outline of the method
The general idea is to adapt the “extended Voronoi region / influence region” method:
1 Assume that your base point lies in a certain domain T0, that we call a “fundamental subdomain”. 2 Compute an extended Voronoi region of D, ExtVorG (D); this is any region guaranteed to contain VorGp(p) for every p ∈ D. 3 Find all other fundamental subdomains D′ (of all types, if D does not contain a fundamental domain of N (G)) such that ExtVorG (D) ∩ ′ ExtVorG (D ) is not empty. Call InflG (D) the union of them.
Lemma For every p ∈ D, all neighbors (i.e., facet producing orbit points) are in InflG (D).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Step 1: the fundamental subdomain(s)
We start with a fundamental domain of N (P213) (remember that P213 ≤ G ≤N (P213)) which is a quarter of a permutahedron.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Step 1: the fundamental subdomain(s)
We start with a fundamental domain of N (P213) (remember that P213 ≤ G ≤N (P213)) which is a quarter of a permutahedron. We subdivide it into four fundamental subdomains A0, B0, C0 and D0.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Step 1: the fundamental subdomain(s)
We start with a fundamental domain of N (P213) (remember that P213 ≤ G ≤N (P213)) which is a quarter of a permutahedron. We subdivide it into four fundamental subdomains A0, B0, C0 and D0.
We consider R3 tiled by these four prototiles via the action of N (P213). In the rest, all extended Voronoi regions and influence regions will be computed as unions of such tiles.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Step 1: the fundamental subdomain(s)
We start with a fundamental domain of N (P213) (remember that P213 ≤ G ≤N (P213)) which is a quarter of a permutahedron. We subdivide it into four fundamental subdomains A0, B0, C0 and D0.
We consider R3 tiled by these four prototiles via the action of N (P213). In the rest, all extended Voronoi regions and influence regions will be computed as unions of such tiles. Call this the “ambient tiling”.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Step 2: the extended Voronoi region(s)
We start with an initial population P of (more than 3000) fundamental subdomains, whose union is guaranteed to contain VorExtG (D) for D ∈ {A0, B0, C0, D0}.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Step 2: the extended Voronoi region(s)
We start with an initial population P of (more than 3000) fundamental subdomains, whose union is guaranteed to contain VorExtG (D) for D ∈ {A0, B0, C0, D0}. We also consider a list of (between 40 and 160) motions from G that are used to “cut out” the extended Voronoi regions.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Step 2: the extended Voronoi region(s)
We start with an initial population P of (more than 3000) fundamental subdomains, whose union is guaranteed to contain VorExtG (D) for D ∈ {A0, B0, C0, D0}. We also consider a list of (between 40 and 160) motions from G that are used to “cut out” the extended Voronoi regions. For each of the four choices of D, the 3000 choices of D′ and choice of motion ρ, we check whether ρ excludes D′ from intersecting ExtVorG (D).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Step 2: the extended Voronoi region(s)
We start with an initial population P of (more than 3000) fundamental subdomains, whose union is guaranteed to contain VorExtG (D) for D ∈ {A0, B0, C0, D0}. We also consider a list of (between 40 and 160) motions from G that are used to “cut out” the extended Voronoi regions. For each of the four choices of D, the 3000 choices of D′ and choice of motion ρ, we check whether ρ excludes D′ from ′ intersecting ExtVorG (D). The non-excluded D form our extended Voronoi region ExtVorG (D) (one for each of {A0, B0, C0, D0}).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups
Table 8 Transformations that we use in each quarter group
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups
Table 9 AsetS3 of triad rotations in Q and, therefore, in all the quarter groups
Table 12 AsetS6 ⊂ S5 of diad rotations parallel to the diagonal of the faces of the unit cube that appear in the group P4132
Table 10 41 2 AsetS4 of diad rotations parallel to the coordinate axes that appear in the groups I g 3 d , I 4132, 2 ! I 43d, I g 3andI 2 3
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Step 3: the influence region(s)
Here we use a feature of our encoding; each tile D′ of the initial population is encoded as the element g ∈N (P213) giving D′ from one of {A0, B0, C0, D0} (plus a label saying which type of tile D′ is).
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Step 3: the influence region(s)
Here we use a feature of our encoding; each tile D′ of the initial population is encoded as the element g ∈N (P213) giving D′ from one of {A0, B0, C0, D0} (plus a label saying which type of tile D′ is). This implies that we can get the influence region simply as
−1 InflG (D) := {g1 ◦ g2 : g1, g2 ∈ ExtVor(D)}
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Step 3: the influence region(s)
Here we use a feature of our encoding; each tile D′ of the initial population is encoded as the element g ∈N (P213) giving D′ from one of {A0, B0, C0, D0} (plus a label saying which type of tile D′ is). This implies that we can get the influence region simply as
−1 InflG (D) := {g1 ◦ g2 : g1, g2 ∈ ExtVor(D)}
Well..., this would be true if we had not several types of “fundamental subdomains”.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Step 3: the influence region(s)
Here we use a feature of our encoding; each tile D′ of the initial population is encoded as the element g ∈N (P213) giving D′ from one of {A0, B0, C0, D0} (plus a label saying which type of tile D′ is). This implies that we can get the influence region simply as
−1 InflG (D) := {g1 ◦ g2 : g1, g2 ∈ ExtVor(D)}
Well..., this would be true if we had not several types of “fundamental subdomains”. Taking the types into account, the formula is rather
−1 ′ InflG (D) = ∪D ∈{A0,B0,C0,D0}{g1 ◦ g2 : ′ g1 ∈ ExtVor(D), g2 ∈ ExtVor(D ) and g1, g2 of the same type}
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups The final results
|G : Q| Aspects Group Our bounds
(1) (2) (3) (4) Final
41 2 8 48 N (Q) = I g 3 d 519 155 100 68 68 4 24 I 4132 264 96 55 55 I 43d 257 78 76 76 2 I g 3260775757 2 24 P4132 135 92 92 ! 12 I 2 3131484646 21 24 P a 3132 8686 1 12 Q = P21369 69 (1) Bounds after processing triad rotations (2) Bounds after diad rotations with axes parallel to the coordinate axes (3) Bounds after diagonal diad rotations (4) Bounds after intersecting with planar projections
F. Santos Number of facets of 3-d Dirichlet stereohedra
Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Conclusions
We have shown that Dirichlet stereohedra in 3-d cannot have more than 92 facets.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Conclusions
We have shown that Dirichlet stereohedra in 3-d cannot have more than 92 facets. Our bound is “group by group”, and it exceeds: 38 in only 21 + 8 groups. 50 in only 9 + 7 groups. 70 in only 4 + 3 groups. 21 80 in only two groups (P4132 and P a 3.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Conclusions
We have shown that Dirichlet stereohedra in 3-d cannot have more than 92 facets. Our bound is “group by group”, and it exceeds: 38 in only 21 + 8 groups. 50 in only 9 + 7 groups. 70 in only 4 + 3 groups. 21 80 in only two groups (P4132 and P a 3. In the classes where our bound is big (non-cubic groups wo. reflections and quarter cubic groups) stereohedra with > 30 facets exist.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Conclusions
We have shown that Dirichlet stereohedra in 3-d cannot have more than 92 facets. Our bound is “group by group”, and it exceeds: 38 in only 21 + 8 groups. 50 in only 9 + 7 groups. 70 in only 4 + 3 groups. 21 80 in only two groups (P4132 and P a 3. In the classes where our bound is big (non-cubic groups wo. reflections and quarter cubic groups) stereohedra with > 30 facets exist. Our bound is 55 for the group I 4132 producing Engel’s 38-faceted stereohedra.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Conclusions
And last but perhaps not least
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Conclusions
And last but perhaps not least Our influence regions (encoded as sets of transformations) can be used as preprocessing for further computations. For each “bad group” we have a list of < 100 explicit transformations that are guaranteed to produce all the facet-defining orbit points for each p in the chosen “fundamental subdomain”.
F. Santos Number of facets of 3-d Dirichlet stereohedra Intro Groups with reflections Non-cubic groups Cubic groups Quarter cubic groups
Quarter groups Conclusions
And last but perhaps not least Our influence regions (encoded as sets of transformations) can be used as preprocessing for further computations. For each “bad group” we have a list of < 100 explicit transformations that are guaranteed to produce all the facet-defining orbit points for each p in the chosen “fundamental subdomain”.
THANK YOU
F. Santos Number of facets of 3-d Dirichlet stereohedra