Math 1230, Notes 6
Sep. 9 , 2014
Math 1230, Notes 6 Sep. 9 , 2014 1 / 1 Does the set of reflections of a triangle form a group by itself?
Does the set of reflections of a tetrahedron form a group by itself?
Glue two copies of a regular tetrahedron together so that they have a face in common, and work out all the rotational symmetries of this new solid. Compare with an octahedron.
Questions to think about.
Math 1230, Notes 6 Sep. 9 , 2014 2 / 1 Does the set of reflections of a tetrahedron form a group by itself?
Glue two copies of a regular tetrahedron together so that they have a face in common, and work out all the rotational symmetries of this new solid. Compare with an octahedron.
Questions to think about.
Does the set of reflections of a triangle form a group by itself?
Math 1230, Notes 6 Sep. 9 , 2014 2 / 1 Glue two copies of a regular tetrahedron together so that they have a face in common, and work out all the rotational symmetries of this new solid. Compare with an octahedron.
Questions to think about.
Does the set of reflections of a triangle form a group by itself?
Does the set of reflections of a tetrahedron form a group by itself?
Math 1230, Notes 6 Sep. 9 , 2014 2 / 1 Questions to think about.
Does the set of reflections of a triangle form a group by itself?
Does the set of reflections of a tetrahedron form a group by itself?
Glue two copies of a regular tetrahedron together so that they have a face in common, and work out all the rotational symmetries of this new solid. Compare with an octahedron.
Math 1230, Notes 6 Sep. 9 , 2014 2 / 1 Definition: A crystal is a solid in which the atoms form a periodic structure.
SYMMETRY AND ITS RELATION TO CRYSTALLOGRAPHY
Math 1230, Notes 6 Sep. 9 , 2014 3 / 1 SYMMETRY AND ITS RELATION TO CRYSTALLOGRAPHY
Definition: A crystal is a solid in which the atoms form a periodic structure.
Math 1230, Notes 6 Sep. 9 , 2014 3 / 1 Different kinds found in crystals: translational, rotational, reflectional
A lattice of points form a periodic structure” if and only if it exhibits translational symmetry. :
Key concept for studying crystals:
SYMMETRY
Math 1230, Notes 6 Sep. 9 , 2014 4 / 1 A lattice of points form a periodic structure” if and only if it exhibits translational symmetry. :
Key concept for studying crystals:
SYMMETRY Different kinds found in crystals: translational, rotational, reflectional
Math 1230, Notes 6 Sep. 9 , 2014 4 / 1 Key concept for studying crystals:
SYMMETRY Different kinds found in crystals: translational, rotational, reflectional
A lattice of points form a periodic structure” if and only if it exhibits translational symmetry. :
Math 1230, Notes 6 Sep. 9 , 2014 4 / 1 translational symmetry (all points move)
Math 1230, Notes 6 Sep. 9 , 2014 5 / 1 translational symmetry (all points move)
Math 1230, Notes 6 Sep. 9 , 2014 6 / 1 translational symmetry (all points move)
Math 1230, Notes 6 Sep. 9 , 2014 7 / 1 translational symmetry (all points move)
Math 1230, Notes 6 Sep. 9 , 2014 8 / 1 Rotational Symmetry
6 fold rotational symmetry around a fixed point.
Math 1230, Notes 6 Sep. 9 , 2014 9 / 1 Rotational Symmetry
6 fold rotational symmetry around a fixed point.
Math 1230, Notes 6 Sep. 9 , 2014 10 / 1 Rotational Symmetry
6 fold rotational symmetry around a fixed point.
Math 1230, Notes 6 Sep. 9 , 2014 11 / 1 Rotational Symmetry
6 fold rotational symmetry around a fixed point.
Math 1230, Notes 6 Sep. 9 , 2014 12 / 1 Rotational Symmetry
6 fold rotational symmetry around a fixed point.
Math 1230, Notes 6 Sep. 9 , 2014 13 / 1 Rotational Symmetry
6 fold rotational symmetry around a fixed point.
Math 1230, Notes 6 Sep. 9 , 2014 14 / 1 Rotational Symmetry
6 fold rotational symmetry around a fixed point.
Math 1230, Notes 6 Sep. 9 , 2014 15 / 1 Reflectional Symmetry
Reflection across a line
Math 1230, Notes 6 Sep. 9 , 2014 16 / 1 Reflectional Symmetry
Reflection across a line
Math 1230, Notes 6 Sep. 9 , 2014 17 / 1 Reflectional Symmetry
Reflection across a line
Math 1230, Notes 6 Sep. 9 , 2014 18 / 1 Reflection through a point
Math 1230, Notes 6 Sep. 9 , 2014 19 / 1 Reflection through a point
Math 1230, Notes 6 Sep. 9 , 2014 20 / 1 Reflection through a point
Math 1230, Notes 6 Sep. 9 , 2014 21 / 1 Reflection through a point
Math 1230, Notes 6 Sep. 9 , 2014 22 / 1 Reflection through a point
Math 1230, Notes 6 Sep. 9 , 2014 23 / 1 Applications to Crystallography Definition: A crystal is a substance in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating three-dimensional pattern. (From “About.com Chemistry”)
Math 1230, Notes 6 Sep. 9 , 2014 24 / 1 Theorem: Suppose that a rotation by an angle α leaves the lattice of 2π points unchanged. Then α must be n for some n ∈ {3, 4, 6} .
In other words, a crystal can only have 3, 4, or 6 fold rotational symmetry.
For example, a lattice cannot have five-fold rotational symmetry.
(Coxeter, An Introduction to Geometry, 1989)
Corollary: Dodecahedra and Icosahedra cannot be part of a crystal.
Crystallographic Restriction Theorem:
Definition: A lattice is an infinite set of points in Rn such that for some non-zero vector v, translation of all the points by v leaves the set of points unchanged.
Math 1230, Notes 6 Sep. 9 , 2014 25 / 1 (Coxeter, An Introduction to Geometry, 1989)
Corollary: Dodecahedra and Icosahedra cannot be part of a crystal.
Crystallographic Restriction Theorem:
Definition: A lattice is an infinite set of points in Rn such that for some non-zero vector v, translation of all the points by v leaves the set of points unchanged.
Theorem: Suppose that a rotation by an angle α leaves the lattice of 2π points unchanged. Then α must be n for some n ∈ {3, 4, 6} .
In other words, a crystal can only have 3, 4, or 6 fold rotational symmetry.
For example, a lattice cannot have five-fold rotational symmetry.
Math 1230, Notes 6 Sep. 9 , 2014 25 / 1 Corollary: Dodecahedra and Icosahedra cannot be part of a crystal.
Crystallographic Restriction Theorem:
Definition: A lattice is an infinite set of points in Rn such that for some non-zero vector v, translation of all the points by v leaves the set of points unchanged.
Theorem: Suppose that a rotation by an angle α leaves the lattice of 2π points unchanged. Then α must be n for some n ∈ {3, 4, 6} .
In other words, a crystal can only have 3, 4, or 6 fold rotational symmetry.
For example, a lattice cannot have five-fold rotational symmetry.
(Coxeter, An Introduction to Geometry, 1989)
Math 1230, Notes 6 Sep. 9 , 2014 25 / 1 Crystallographic Restriction Theorem:
Definition: A lattice is an infinite set of points in Rn such that for some non-zero vector v, translation of all the points by v leaves the set of points unchanged.
Theorem: Suppose that a rotation by an angle α leaves the lattice of 2π points unchanged. Then α must be n for some n ∈ {3, 4, 6} .
In other words, a crystal can only have 3, 4, or 6 fold rotational symmetry.
For example, a lattice cannot have five-fold rotational symmetry.
(Coxeter, An Introduction to Geometry, 1989)
Corollary: Dodecahedra and Icosahedra cannot be part of a crystal.
Math 1230, Notes 6 Sep. 9 , 2014 25 / 1 Proof: We can assume that 0 < α < 2π. If a rotation by α leaves the lattice unchanged, then rotations by 2α, 3α, etc. also leave it unchanged. In particular, let the rotations be around p, and rotate the point q = p + v. Let ρ be the rotation around p. The points q, rq, r 2q, etc. all lie on a circle around p. 2π n Case (1); α = n for some positive integer n. Then r q = q. The points q, rq, ..., r n−1q are vertices of a regular n -gon. By standard Euclidean geometry we find that sum of the interior angles (all equal) of this n - gon is (n − 2) π. So each individual interior angle is (n−2)π n . On the other hand, treating the edges as vectors shows that 2π interior angle must be some fraction m . But the only values of n and m which can satisfy this are n = 3, m = 6, n = 4, m = 4 and n = 6, m = 3.
Theorem: Suppose that a rotation by an angle α leaves the lattice of points unchanged. Choose the smallest possible α for which this is 2π true. Then α must be n for some n ∈ {3, 4, 6} .
Math 1230, Notes 6 Sep. 9 , 2014 26 / 1 2π n Case (1); α = n for some positive integer n. Then r q = q. The points q, rq, ..., r n−1q are vertices of a regular n -gon. By standard Euclidean geometry we find that sum of the interior angles (all equal) of this n - gon is (n − 2) π. So each individual interior angle is (n−2)π n . On the other hand, treating the edges as vectors shows that 2π interior angle must be some fraction m . But the only values of n and m which can satisfy this are n = 3, m = 6, n = 4, m = 4 and n = 6, m = 3.
Theorem: Suppose that a rotation by an angle α leaves the lattice of points unchanged. Choose the smallest possible α for which this is 2π true. Then α must be n for some n ∈ {3, 4, 6} . Proof: We can assume that 0 < α < 2π. If a rotation by α leaves the lattice unchanged, then rotations by 2α, 3α, etc. also leave it unchanged. In particular, let the rotations be around p, and rotate the point q = p + v. Let ρ be the rotation around p. The points q, rq, r 2q, etc. all lie on a circle around p.
Math 1230, Notes 6 Sep. 9 , 2014 26 / 1 Theorem: Suppose that a rotation by an angle α leaves the lattice of points unchanged. Choose the smallest possible α for which this is 2π true. Then α must be n for some n ∈ {3, 4, 6} . Proof: We can assume that 0 < α < 2π. If a rotation by α leaves the lattice unchanged, then rotations by 2α, 3α, etc. also leave it unchanged. In particular, let the rotations be around p, and rotate the point q = p + v. Let ρ be the rotation around p. The points q, rq, r 2q, etc. all lie on a circle around p. 2π n Case (1); α = n for some positive integer n. Then r q = q. The points q, rq, ..., r n−1q are vertices of a regular n -gon. By standard Euclidean geometry we find that sum of the interior angles (all equal) of this n - gon is (n − 2) π. So each individual interior angle is (n−2)π n . On the other hand, treating the edges as vectors shows that 2π interior angle must be some fraction m . But the only values of n and m which can satisfy this are n = 3, m = 6, n = 4, m = 4 and n = 6, m = 3.
Math 1230, Notes 6 Sep. 9 , 2014 26 / 1 2π Case (2) α 6= n for any n. In this case, some number of rotations, say mr, comes to a point between q and rq on the circle. This point is closer to q than is rq. But this means that some smaller rotation leaves the lattice unchanged, a contradiction.
Math 1230, Notes 6 Sep. 9 , 2014 27 / 1 Dan Schactman, of the Israel Institute of Technology, muttered to himself: “Eyn chaya kazo” – There can be no such creature.
In 1984 the world of crystallography was shocked by the following picture:
Math 1230, Notes 6 Sep. 9 , 2014 28 / 1 Dan Schactman, of the Israel Institute of Technology, muttered to himself: “Eyn chaya kazo” – There can be no such creature.
In 1984 the world of crystallography was shocked by the following picture:
Math 1230, Notes 6 Sep. 9 , 2014 28 / 1 In 1984 the world of crystallography was shocked by the following picture:
Dan Schactman, of the Israel Institute of Technology, muttered to himself: “Eyn chaya kazo” – There can be no such creature.
Math 1230, Notes 6 Sep. 9 , 2014 28 / 1 But eventually his paper began: “We have observed a metallic solid (Al-14-at.%-Mn) with long-range orientational order, but with icosahedral point group symmetry, which is inconsistent with lattice translations.”
Math 1230, Notes 6 Sep. 9 , 2014 29 / 1 From Wikipedia: Linus Pauling is noted as saying ”There is no such thing as quasicrystals, only quasi-scientists.” [Reuters] ... The head of Shechtman’s research group told him to ”go back and read the textbook” and a couple of days later ”asked him to leave for ’bringing disgrace’ on the team.” [The Guardian]
Pauling tried to show that an alternative phenomenon, called “twinning” explained the picture, but he made some computational errors, and when they were fixed his argument collapsed.
Nevertheless, he apparently went to his grave unconvinced.
Very Controversial!
Math 1230, Notes 6 Sep. 9 , 2014 30 / 1 Pauling tried to show that an alternative phenomenon, called “twinning” explained the picture, but he made some computational errors, and when they were fixed his argument collapsed.
Nevertheless, he apparently went to his grave unconvinced.
Very Controversial!
From Wikipedia: Linus Pauling is noted as saying ”There is no such thing as quasicrystals, only quasi-scientists.” [Reuters] ... The head of Shechtman’s research group told him to ”go back and read the textbook” and a couple of days later ”asked him to leave for ’bringing disgrace’ on the team.” [The Guardian]
Math 1230, Notes 6 Sep. 9 , 2014 30 / 1 Nevertheless, he apparently went to his grave unconvinced.
Very Controversial!
From Wikipedia: Linus Pauling is noted as saying ”There is no such thing as quasicrystals, only quasi-scientists.” [Reuters] ... The head of Shechtman’s research group told him to ”go back and read the textbook” and a couple of days later ”asked him to leave for ’bringing disgrace’ on the team.” [The Guardian]
Pauling tried to show that an alternative phenomenon, called “twinning” explained the picture, but he made some computational errors, and when they were fixed his argument collapsed.
Math 1230, Notes 6 Sep. 9 , 2014 30 / 1 Very Controversial!
From Wikipedia: Linus Pauling is noted as saying ”There is no such thing as quasicrystals, only quasi-scientists.” [Reuters] ... The head of Shechtman’s research group told him to ”go back and read the textbook” and a couple of days later ”asked him to leave for ’bringing disgrace’ on the team.” [The Guardian]
Pauling tried to show that an alternative phenomenon, called “twinning” explained the picture, but he made some computational errors, and when they were fixed his argument collapsed.
Nevertheless, he apparently went to his grave unconvinced.
Math 1230, Notes 6 Sep. 9 , 2014 30 / 1 A mathematician may ask: What is the explanation for what looks like a material with large scale “order” but which cannot have its atoms or molecules arranged periodically?
Math 1230, Notes 6 Sep. 9 , 2014 31 / 1 Aristotle. (Plato’s student, 384-322 BC)
“It is agreed that there are only three (regular) plane figures which can fill a space, the (equilateral) triangle, the square, and the (regular) hexagon, .... ”
tilings by regular polygons
Tiling in two dimensions has a long history.
Math 1230, Notes 6 Sep. 9 , 2014 32 / 1 “It is agreed that there are only three (regular) plane figures which can fill a space, the (equilateral) triangle, the square, and the (regular) hexagon, .... ”
tilings by regular polygons
Tiling in two dimensions has a long history.
Aristotle. (Plato’s student, 384-322 BC)
Math 1230, Notes 6 Sep. 9 , 2014 32 / 1 tilings by regular polygons
Tiling in two dimensions has a long history.
Aristotle. (Plato’s student, 384-322 BC)
“It is agreed that there are only three (regular) plane figures which can fill a space, the (equilateral) triangle, the square, and the (regular) hexagon, .... ”
Math 1230, Notes 6 Sep. 9 , 2014 32 / 1 Tiling in two dimensions has a long history.
Aristotle. (Plato’s student, 384-322 BC)
“It is agreed that there are only three (regular) plane figures which can fill a space, the (equilateral) triangle, the square, and the (regular) hexagon, .... ”
tilings by regular polygons
Math 1230, Notes 6 Sep. 9 , 2014 32 / 1 Math 1230, Notes 6 Sep. 9 , 2014 33 / 1 Johannes Kepler, in 1619, found all tilings by combinations of regular polygons:
Math 1230, Notes 6 Sep. 9 , 2014 34 / 1 Johannes Kepler, in 1619, found all tilings by combinations of regular polygons:
Math 1230, Notes 6 Sep. 9 , 2014 34 / 1 but rearranging the same tiles give periodicity.
Here is an aperiodic tiling:
These can all be shown to be periodic, and have a translational symmetry.
Math 1230, Notes 6 Sep. 9 , 2014 35 / 1 but rearranging the same tiles give periodicity.
These can all be shown to be periodic, and have a translational symmetry. Here is an aperiodic tiling:
Math 1230, Notes 6 Sep. 9 , 2014 35 / 1 These can all be shown to be periodic, and have a translational symmetry. Here is an aperiodic tiling:
but rearranging the same tiles give periodicity.
Math 1230, Notes 6 Sep. 9 , 2014 35 / 1 Robert Berger in 1966: yes, there is a set. It has 20,426 different tiles.
R. Penrose (1974) Yes, there is such a set. It has two different tiles.
Question (Hao Wang, 1961): Is there a (finite) set of tiles such that: (a) they tile the plane in at least one way (b) every possible tiling with these tiles is aperiodic ?
Math 1230, Notes 6 Sep. 9 , 2014 36 / 1 R. Penrose (1974) Yes, there is such a set. It has two different tiles.
Question (Hao Wang, 1961): Is there a (finite) set of tiles such that: (a) they tile the plane in at least one way (b) every possible tiling with these tiles is aperiodic ?
Robert Berger in 1966: yes, there is a set. It has 20,426 different tiles.
Math 1230, Notes 6 Sep. 9 , 2014 36 / 1 Question (Hao Wang, 1961): Is there a (finite) set of tiles such that: (a) they tile the plane in at least one way (b) every possible tiling with these tiles is aperiodic ?
Robert Berger in 1966: yes, there is a set. It has 20,426 different tiles.
R. Penrose (1974) Yes, there is such a set. It has two different tiles.
Math 1230, Notes 6 Sep. 9 , 2014 36 / 1 Math 1230, Notes 6 Sep. 9 , 2014 37 / 1 arrows must match Problem: Tile the plane with these. demo
Math 1230, Notes 6 Sep. 9 , 2014 37 / 1 Problem: Tile the plane with these. demo
arrows must match
Math 1230, Notes 6 Sep. 9 , 2014 37 / 1 demo
arrows must match Problem: Tile the plane with these.
Math 1230, Notes 6 Sep. 9 , 2014 37 / 1 arrows must match Problem: Tile the plane with these. demo
Math 1230, Notes 6 Sep. 9 , 2014 37 / 1 Note the 5-fold symmetry, so this cannot extend periodically.
Math 1230, Notes 6 Sep. 9 , 2014 38 / 1 In 1993 the International Union of Crystallography changed the definition of crystal from “a substance in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating three-dimensional pattern” to “any solid having an essentially discrete diffraction diagram.”
Is this important? The accepted theoretical explanation for the existence of quasi-crystal is based on Penrose tiles.
Math 1230, Notes 6 Sep. 9 , 2014 39 / 1 Is this important? The accepted theoretical explanation for the existence of quasi-crystal is based on Penrose tiles.
In 1993 the International Union of Crystallography changed the definition of crystal from “a substance in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating three-dimensional pattern” to “any solid having an essentially discrete diffraction diagram.”
Math 1230, Notes 6 Sep. 9 , 2014 39 / 1 Applications: verdict still out; (surgical instruments, LED lights, non-stick frying pans, ... ? )
In 2009 a natural quasi-crystal was found in Russia. It is believed to be from a meteorite.
Math 1230, Notes 6 Sep. 9 , 2014 40 / 1 In 2009 a natural quasi-crystal was found in Russia. It is believed to be from a meteorite. Applications: verdict still out; (surgical instruments, LED lights, non-stick frying pans, ... ? )
Math 1230, Notes 6 Sep. 9 , 2014 40 / 1 In 2011 Dan Shechtman won the Nobel Prize for Chemistry
Math 1230, Notes 6 Sep. 9 , 2014 41 / 1 Aristotle: “It is agreed that there are only three (regular) plane figures which can fill a space, the (equilateral) triangle, the square, and the (regular) hexagon, and only two regular solids, the tetrahedron and the cube. ”
Math 1230, Notes 6 Sep. 9 , 2014 42 / 1 Math 1230, Notes 6 Sep. 9 , 2014 43 / 1 Voronoi diagram: 2 3 Consider a finite or infinite set of points p1, ..., pn, .. in R or R . The Voronoi diagram is a subdivision of Rn into subsets of the following type:
n Vj = {x ∈ R | |x − pj | ≤ |x − pk | for all k 6= j}
In other words, Vj is the set of all points which are at least as close to pj as to any other of the pk .
Of particular interest: the rhombic dodecahdron. dual of an Archimedean solid. (The cubeoctohedron) – discovered by Kepler (1571-1630),
Math 1230, Notes 6 Sep. 9 , 2014 44 / 1 In other words, Vj is the set of all points which are at least as close to pj as to any other of the pk .
Of particular interest: the rhombic dodecahdron. dual of an Archimedean solid. (The cubeoctohedron) – discovered by Kepler (1571-1630),
Voronoi diagram: 2 3 Consider a finite or infinite set of points p1, ..., pn, .. in R or R . The Voronoi diagram is a subdivision of Rn into subsets of the following type:
n Vj = {x ∈ R | |x − pj | ≤ |x − pk | for all k 6= j}
Math 1230, Notes 6 Sep. 9 , 2014 44 / 1 Of particular interest: the rhombic dodecahdron. dual of an Archimedean solid. (The cubeoctohedron) – discovered by Kepler (1571-1630),
Voronoi diagram: 2 3 Consider a finite or infinite set of points p1, ..., pn, .. in R or R . The Voronoi diagram is a subdivision of Rn into subsets of the following type:
n Vj = {x ∈ R | |x − pj | ≤ |x − pk | for all k 6= j}
In other words, Vj is the set of all points which are at least as close to pj as to any other of the pk .
Math 1230, Notes 6 Sep. 9 , 2014 44 / 1 2. Start with the set of all points (m, n) with integer coefficients. These are then the vertices of a tessellation of R2 into unit squares with horizontal and vertical edges. Add to this set all points 1 1 m + 2 , n and m, n + 2 , which is to say add the centers of all the edges of the previous squares. Find the resulting Voronoi diagram. (Class exercise)
Examples:
1. pi = (i, 0) , for i = 1, .., ... 1 1 Then for each i, Vj is the vertical strip i − 2 ≤ x ≤ i + 2 .
Math 1230, Notes 6 Sep. 9 , 2014 45 / 1 Examples:
1. pi = (i, 0) , for i = 1, .., ... 1 1 Then for each i, Vj is the vertical strip i − 2 ≤ x ≤ i + 2 .
2. Start with the set of all points (m, n) with integer coefficients. These are then the vertices of a tessellation of R2 into unit squares with horizontal and vertical edges. Add to this set all points 1 1 m + 2 , n and m, n + 2 , which is to say add the centers of all the edges of the previous squares. Find the resulting Voronoi diagram. (Class exercise)
Math 1230, Notes 6 Sep. 9 , 2014 45 / 1 3. Now, in three dimensions do something similar. Start with the set of all vertices (m, n, o) where m, n, o are integers. This tessellates R3 into cubes. Now add the centers of all the faces of these cubes (not the centers of the edges, or the center of the whole cube). It turns out that the resulting Voronoi diagram tessellates R3 with rhombic dodecahedra!
Math 1230, Notes 6 Sep. 9 , 2014 46 / 1 3. Now, in three dimensions do something similar. Start with the set of all vertices (m, n, o) where m, n, o are integers. This tessellates R3 into cubes. Now add the centers of all the faces of these cubes (not the centers of the edges, or the center of the whole cube). It turns out that the resulting Voronoi diagram tessellates R3 with rhombic dodecahedra!
Math 1230, Notes 6 Sep. 9 , 2014 46 / 1 Voronoi diagrams have many applications – a 2009 book, “Spatial Tessellations: Concepts and Applications of Voronoi Diagrams” already has over 4000 citations. The emphasis here is on spatial data analysis.
Math 1230, Notes 6 Sep. 9 , 2014 47 / 1 Time line: around 350 bc. Aristotle asserted that tetrahedra can fill space. 6th century: Simplicius stated that twelve congruent tetrahedra can touch at a point, and they will fill the space around this point. around 1150: Abu al-Walid Mohammed ibn Ahmad al Rushid reasserted this claim ? How would anyone come up with this idea, of 12 tetrahedra meeting at a point?
Aristotle was wrong! The tetrahedron does not fill (tile) space.
Math 1230, Notes 6 Sep. 9 , 2014 48 / 1 around 350 bc. Aristotle asserted that tetrahedra can fill space. 6th century: Simplicius stated that twelve congruent tetrahedra can touch at a point, and they will fill the space around this point. around 1150: Abu al-Walid Mohammed ibn Ahmad al Rushid reasserted this claim ? How would anyone come up with this idea, of 12 tetrahedra meeting at a point?
Aristotle was wrong! The tetrahedron does not fill (tile) space.
Time line:
Math 1230, Notes 6 Sep. 9 , 2014 48 / 1 6th century: Simplicius stated that twelve congruent tetrahedra can touch at a point, and they will fill the space around this point. around 1150: Abu al-Walid Mohammed ibn Ahmad al Rushid reasserted this claim ? How would anyone come up with this idea, of 12 tetrahedra meeting at a point?
Aristotle was wrong! The tetrahedron does not fill (tile) space.
Time line: around 350 bc. Aristotle asserted that tetrahedra can fill space.
Math 1230, Notes 6 Sep. 9 , 2014 48 / 1 around 1150: Abu al-Walid Mohammed ibn Ahmad al Rushid reasserted this claim ? How would anyone come up with this idea, of 12 tetrahedra meeting at a point?
Aristotle was wrong! The tetrahedron does not fill (tile) space.
Time line: around 350 bc. Aristotle asserted that tetrahedra can fill space. 6th century: Simplicius stated that twelve congruent tetrahedra can touch at a point, and they will fill the space around this point.
Math 1230, Notes 6 Sep. 9 , 2014 48 / 1 ? How would anyone come up with this idea, of 12 tetrahedra meeting at a point?
Aristotle was wrong! The tetrahedron does not fill (tile) space.
Time line: around 350 bc. Aristotle asserted that tetrahedra can fill space. 6th century: Simplicius stated that twelve congruent tetrahedra can touch at a point, and they will fill the space around this point. around 1150: Abu al-Walid Mohammed ibn Ahmad al Rushid reasserted this claim
Math 1230, Notes 6 Sep. 9 , 2014 48 / 1 Aristotle was wrong! The tetrahedron does not fill (tile) space.
Time line: around 350 bc. Aristotle asserted that tetrahedra can fill space. 6th century: Simplicius stated that twelve congruent tetrahedra can touch at a point, and they will fill the space around this point. around 1150: Abu al-Walid Mohammed ibn Ahmad al Rushid reasserted this claim ? How would anyone come up with this idea, of 12 tetrahedra meeting at a point?
Math 1230, Notes 6 Sep. 9 , 2014 48 / 1 1266: Roger Bacon wrote that there is fool in Paris who says that 12 is wrong, and 20 congruent tetrahedra can meet at a single point, where they fill the space. But Bacon added that one can’t be absolutely sure that 12 is right without a proof in the manner of Euclid. ? why 20 ?
1274: Thomas Aquinas wrote lectures on Aristotle’s work but these remained incomplete at his death. 1320: Bradwardinus, later Archbishop of Canterbury, stated that there were different opinions about wheth research area, with material scientists, physicists and mathematicians all studying tetrahedral packings. 1480: Paul of Middleburg showed that tetrahedra cannot fill 100% of space.
1217: Michael Scotus translated this from the Arabic into Latin.
Math 1230, Notes 6 Sep. 9 , 2014 49 / 1 ? why 20 ?
1274: Thomas Aquinas wrote lectures on Aristotle’s work but these remained incomplete at his death. 1320: Bradwardinus, later Archbishop of Canterbury, stated that there were different opinions about wheth research area, with material scientists, physicists and mathematicians all studying tetrahedral packings. 1480: Paul of Middleburg showed that tetrahedra cannot fill 100% of space.
1217: Michael Scotus translated this from the Arabic into Latin. 1266: Roger Bacon wrote that there is fool in Paris who says that 12 is wrong, and 20 congruent tetrahedra can meet at a single point, where they fill the space. But Bacon added that one can’t be absolutely sure that 12 is right without a proof in the manner of Euclid.
Math 1230, Notes 6 Sep. 9 , 2014 49 / 1 1274: Thomas Aquinas wrote lectures on Aristotle’s work but these remained incomplete at his death. 1320: Bradwardinus, later Archbishop of Canterbury, stated that there were different opinions about wheth research area, with material scientists, physicists and mathematicians all studying tetrahedral packings. 1480: Paul of Middleburg showed that tetrahedra cannot fill 100% of space.
1217: Michael Scotus translated this from the Arabic into Latin. 1266: Roger Bacon wrote that there is fool in Paris who says that 12 is wrong, and 20 congruent tetrahedra can meet at a single point, where they fill the space. But Bacon added that one can’t be absolutely sure that 12 is right without a proof in the manner of Euclid. ? why 20 ?
Math 1230, Notes 6 Sep. 9 , 2014 49 / 1 1320: Bradwardinus, later Archbishop of Canterbury, stated that there were different opinions about wheth research area, with material scientists, physicists and mathematicians all studying tetrahedral packings. 1480: Paul of Middleburg showed that tetrahedra cannot fill 100% of space.
1217: Michael Scotus translated this from the Arabic into Latin. 1266: Roger Bacon wrote that there is fool in Paris who says that 12 is wrong, and 20 congruent tetrahedra can meet at a single point, where they fill the space. But Bacon added that one can’t be absolutely sure that 12 is right without a proof in the manner of Euclid. ? why 20 ?
1274: Thomas Aquinas wrote lectures on Aristotle’s work but these remained incomplete at his death.
Math 1230, Notes 6 Sep. 9 , 2014 49 / 1 1480: Paul of Middleburg showed that tetrahedra cannot fill 100% of space.
1217: Michael Scotus translated this from the Arabic into Latin. 1266: Roger Bacon wrote that there is fool in Paris who says that 12 is wrong, and 20 congruent tetrahedra can meet at a single point, where they fill the space. But Bacon added that one can’t be absolutely sure that 12 is right without a proof in the manner of Euclid. ? why 20 ?
1274: Thomas Aquinas wrote lectures on Aristotle’s work but these remained incomplete at his death. 1320: Bradwardinus, later Archbishop of Canterbury, stated that there were different opinions about wheth research area, with material scientists, physicists and mathematicians all studying tetrahedral packings.
Math 1230, Notes 6 Sep. 9 , 2014 49 / 1 1217: Michael Scotus translated this from the Arabic into Latin. 1266: Roger Bacon wrote that there is fool in Paris who says that 12 is wrong, and 20 congruent tetrahedra can meet at a single point, where they fill the space. But Bacon added that one can’t be absolutely sure that 12 is right without a proof in the manner of Euclid. ? why 20 ?
1274: Thomas Aquinas wrote lectures on Aristotle’s work but these remained incomplete at his death. 1320: Bradwardinus, later Archbishop of Canterbury, stated that there were different opinions about wheth research area, with material scientists, physicists and mathematicians all studying tetrahedral packings. 1480: Paul of Middleburg showed that tetrahedra cannot fill 100% of space.
Math 1230, Notes 6 Sep. 9 , 2014 49 / 1 Partial Proof: One can calculate the “dihedral angle” of a tetrahedron (the angle between two faces which meet along an edge). π o 0 This turns out to be α = arccos 3 ≈ 70 32 . If five tetrahedra meet at an edge, there is a gap of about 7o.
Math 1230, Notes 6 Sep. 9 , 2014 50 / 1 Math 1230, Notes 6 Sep. 9 , 2014 51 / 1 1900: David Hilbert’s 23 problems. # 18: “I point out the following question, ... important to number theory and perhaps sometimes useful to physics and chemistry:
How can one arrange most densely in space an infinite number of equal solids of given form, e.g., spheres with given radii or regular tetrahedra with given edges (or in prescribed position); that is, how can one so fit them together that the ratio of the filled to the unfilled space may be as great as possible?”
“Modern” time line 1480: Paul of Middleburg showed that tetrahedra cannot fill 100% of space.
Math 1230, Notes 6 Sep. 9 , 2014 52 / 1 # 18: “I point out the following question, ... important to number theory and perhaps sometimes useful to physics and chemistry:
How can one arrange most densely in space an infinite number of equal solids of given form, e.g., spheres with given radii or regular tetrahedra with given edges (or in prescribed position); that is, how can one so fit them together that the ratio of the filled to the unfilled space may be as great as possible?”
“Modern” time line 1480: Paul of Middleburg showed that tetrahedra cannot fill 100% of space.
1900: David Hilbert’s 23 problems.
Math 1230, Notes 6 Sep. 9 , 2014 52 / 1 How can one arrange most densely in space an infinite number of equal solids of given form, e.g., spheres with given radii or regular tetrahedra with given edges (or in prescribed position); that is, how can one so fit them together that the ratio of the filled to the unfilled space may be as great as possible?”
“Modern” time line 1480: Paul of Middleburg showed that tetrahedra cannot fill 100% of space.
1900: David Hilbert’s 23 problems. # 18: “I point out the following question, ... important to number theory and perhaps sometimes useful to physics and chemistry:
Math 1230, Notes 6 Sep. 9 , 2014 52 / 1 “Modern” time line 1480: Paul of Middleburg showed that tetrahedra cannot fill 100% of space.
1900: David Hilbert’s 23 problems. # 18: “I point out the following question, ... important to number theory and perhaps sometimes useful to physics and chemistry:
How can one arrange most densely in space an infinite number of equal solids of given form, e.g., spheres with given radii or regular tetrahedra with given edges (or in prescribed position); that is, how can one so fit them together that the ratio of the filled to the unfilled space may be as great as possible?”
Math 1230, Notes 6 Sep. 9 , 2014 52 / 1 2006: Conway (Princeton): Can fill at least 71% of space with tetrahedra. Conjectured that spheres are better at packing than tetrahedra. 2008 Elizabeth Chen (Grad student at U. Mich.): Conjecture false; tetrahedra can fill at least 77.8% of space
(Sphere case: Kepler (1611) conjectured that spheres can fill no more than about 74% of space; proved by Tom Hales, Pitt, 2005.)
Math 1230, Notes 6 Sep. 9 , 2014 53 / 1 2008 Elizabeth Chen (Grad student at U. Mich.): Conjecture false; tetrahedra can fill at least 77.8% of space
(Sphere case: Kepler (1611) conjectured that spheres can fill no more than about 74% of space; proved by Tom Hales, Pitt, 2005.) 2006: Conway (Princeton): Can fill at least 71% of space with tetrahedra. Conjectured that spheres are better at packing than tetrahedra.
Math 1230, Notes 6 Sep. 9 , 2014 53 / 1 (Sphere case: Kepler (1611) conjectured that spheres can fill no more than about 74% of space; proved by Tom Hales, Pitt, 2005.) 2006: Conway (Princeton): Can fill at least 71% of space with tetrahedra. Conjectured that spheres are better at packing than tetrahedra. 2008 Elizabeth Chen (Grad student at U. Mich.): Conjecture false; tetrahedra can fill at least 77.8% of space
Math 1230, Notes 6 Sep. 9 , 2014 53 / 1 2009-2010. Seven papers successively raised this to 85.6347 %.
Math 1230, Notes 6 Sep. 9 , 2014 54 / 1 2009-2010. Seven papers successively raised this to 85.6347 %.
Math 1230, Notes 6 Sep. 9 , 2014 54 / 1 An ”upper bound”: No one has proved that tetrahedra cannot fill as much as 99.999 999 999 999 999 999 999 999 999% of space!
See papers of Elizabeth Chen (use scholar.google.com) ————–
Math 1230, Notes 6 Sep. 9 , 2014 55 / 1 See papers of Elizabeth Chen (use scholar.google.com) ————– An ”upper bound”: No one has proved that tetrahedra cannot fill as much as 99.999 999 999 999 999 999 999 999 999% of space!
Math 1230, Notes 6 Sep. 9 , 2014 55 / 1 Dense-packing crystal structures of physical tetrahedra, Y Kallus, V Elser - Physical Review E, 2011 - APS Communication: A packing of truncated tetrahedra that nearly fills all of space and its melting properties, Y Jiao, S Torquato - The Journal of chemical physics, 2011 Study of Pervious Concrete in Aggregate Gradation Structure, FR Zhao, ZL Ding - Advanced Materials Research, 2012 , ... Fig.2 Tetrahedral structure ...Fig.3 Octahedral structure .. If the aggregates are made of single pore size, the steadiest structure of its basic unit is tetrahedron packing structure Packing systematics of the silica polymorphs: The role played by OO nonbonded interactions in the compression of quartz, RM Thompson, RT Downs - American Mineralogist, 2010. Changes in the unoccupied tetrahedra are responsible for most of the compression in quartz with pressure, as the volume of the Si tetrahedron decreases by <1% over 10.2 GPa,
Recent articles on tetrahedral packing:
Math 1230, Notes 6 Sep. 9 , 2014 56 / 1 Communication: A packing of truncated tetrahedra that nearly fills all of space and its melting properties, Y Jiao, S Torquato - The Journal of chemical physics, 2011 Study of Pervious Concrete in Aggregate Gradation Structure, FR Zhao, ZL Ding - Advanced Materials Research, 2012 , ... Fig.2 Tetrahedral structure ...Fig.3 Octahedral structure .. If the aggregates are made of single pore size, the steadiest structure of its basic unit is tetrahedron packing structure Packing systematics of the silica polymorphs: The role played by OO nonbonded interactions in the compression of quartz, RM Thompson, RT Downs - American Mineralogist, 2010. Changes in the unoccupied tetrahedra are responsible for most of the compression in quartz with pressure, as the volume of the Si tetrahedron decreases by <1% over 10.2 GPa,
Recent articles on tetrahedral packing: Dense-packing crystal structures of physical tetrahedra, Y Kallus, V Elser - Physical Review E, 2011 - APS
Math 1230, Notes 6 Sep. 9 , 2014 56 / 1 Study of Pervious Concrete in Aggregate Gradation Structure, FR Zhao, ZL Ding - Advanced Materials Research, 2012 , ... Fig.2 Tetrahedral structure ...Fig.3 Octahedral structure .. If the aggregates are made of single pore size, the steadiest structure of its basic unit is tetrahedron packing structure Packing systematics of the silica polymorphs: The role played by OO nonbonded interactions in the compression of quartz, RM Thompson, RT Downs - American Mineralogist, 2010. Changes in the unoccupied tetrahedra are responsible for most of the compression in quartz with pressure, as the volume of the Si tetrahedron decreases by <1% over 10.2 GPa,
Recent articles on tetrahedral packing: Dense-packing crystal structures of physical tetrahedra, Y Kallus, V Elser - Physical Review E, 2011 - APS Communication: A packing of truncated tetrahedra that nearly fills all of space and its melting properties, Y Jiao, S Torquato - The Journal of chemical physics, 2011
Math 1230, Notes 6 Sep. 9 , 2014 56 / 1 Packing systematics of the silica polymorphs: The role played by OO nonbonded interactions in the compression of quartz, RM Thompson, RT Downs - American Mineralogist, 2010. Changes in the unoccupied tetrahedra are responsible for most of the compression in quartz with pressure, as the volume of the Si tetrahedron decreases by <1% over 10.2 GPa,
Recent articles on tetrahedral packing: Dense-packing crystal structures of physical tetrahedra, Y Kallus, V Elser - Physical Review E, 2011 - APS Communication: A packing of truncated tetrahedra that nearly fills all of space and its melting properties, Y Jiao, S Torquato - The Journal of chemical physics, 2011 Study of Pervious Concrete in Aggregate Gradation Structure, FR Zhao, ZL Ding - Advanced Materials Research, 2012 , ... Fig.2 Tetrahedral structure ...Fig.3 Octahedral structure .. If the aggregates are made of single pore size, the steadiest structure of its basic unit is tetrahedron packing structure
Math 1230, Notes 6 Sep. 9 , 2014 56 / 1 Recent articles on tetrahedral packing: Dense-packing crystal structures of physical tetrahedra, Y Kallus, V Elser - Physical Review E, 2011 - APS Communication: A packing of truncated tetrahedra that nearly fills all of space and its melting properties, Y Jiao, S Torquato - The Journal of chemical physics, 2011 Study of Pervious Concrete in Aggregate Gradation Structure, FR Zhao, ZL Ding - Advanced Materials Research, 2012 , ... Fig.2 Tetrahedral structure ...Fig.3 Octahedral structure .. If the aggregates are made of single pore size, the steadiest structure of its basic unit is tetrahedron packing structure Packing systematics of the silica polymorphs: The role played by OO nonbonded interactions in the compression of quartz, RM Thompson, RT Downs - American Mineralogist, 2010. Changes in the unoccupied tetrahedra are responsible for most of the compression in quartz with pressure, as the volume of the Si tetrahedron decreases by <1% over 10.2 GPa,
Math 1230, Notes 6 Sep. 9 , 2014 56 / 1 Illustrates “The unreasonable effectiveness of Mathematics in the Natural Sciences”, by Eugene Wigner, Nobel Laureate in physics, 1963.
Math 1230, Notes 6 Sep. 9 , 2014 57 / 1 Math 1230, Notes 6 Sep. 9 , 2014 58 / 1 Math 1230, Notes 6 Sep. 9 , 2014 59 / 1 Math 1230, Notes 6 Sep. 9 , 2014 60 / 1 Math 1230, Notes 6 Sep. 9 , 2014 61 / 1 Math 1230, Notes 6 Sep. 9 , 2014 62 / 1 Math 1230, Notes 6 Sep. 9 , 2014 63 / 1