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Cube, Cube-Octahedron, Or Rhombododecahedron CUBE,CUBE, CUBE-OCTAHEDRON,CUBE-OCTAHEDRON, OROR RHOMBODODECAHEDRONRHOMBODODECAHEDRON ?? J. SERRA Centre de Morphologie Mathématique, Ecole des Mines de Paris France IWVF3, Capri, May 28-30, 1997 IWVF3, Capri, May 28-30, 1997 Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 1 TheThe ThreeThree CubicCubic SystemsSystems of of Grids Grids (I) (I) PlacePlace a a point point at at the the center center of of a a Euclidean Euclidean unit unit cube. cube. You You cancan project project it: it: 1/1/ onon all all of of the the six six faces;faces; 2/2/ onon all all of of the the eighteight vertices;vertices; 3/3/ on on all all of of the the twelve twelve edges. edges. InIn each each case, case, by by symmetry, symmetry, all all projections projections have have a a same same length. length. SoSo byby translation translation each each systemsystem generatesgenerates a a netnet ofof vertices vertices and and edgesedges through through the the whole whole space, space, that that we we shall shall call call a a grid. grid. Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 2 TheThe ThreeThree CubicCubic SystemsSystems of of Grids Grids (II) (II) InIn these these three three grids, grids, the the families families of of vertices vertices admit admit the the followingfollowing interpretations interpretations : : •• 1rst1rst casecase:: translations translations of of the the unit unit cube cube vertices, vertices, ⇒⇒ CUBICCUBIC GRIDGRID •• 2nd2nd casecase:: the the centres centres of of the the previous previous cubes cubes are are added added to to their their vertices,vertices, ⇒⇒ CENTEREDCENTERED CUBICCUBIC GRIDGRID (( cccc grid grid ) ) •• 3rd3rd casecase:: the the centres centres of of the the faces faces are are added added to to the the vertices vertices of of thethe cubic cubic grid. grid. ⇒⇒ FACE-CENTEREDFACE-CENTERED CUBICCUBIC GRIDGRID (( fcc fcc gridgrid ) ) Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 3 FirstFirst NeighboursNeighbours 8 neighbours 6 neighbours 12 neighbours for the cc grid for the cubic grid for the fcc grid Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 4 CubicCubic GridGrid NeighboursNeighbours Patterns Patterns •• The The sixtsixt firstfirst neighboursneighbours generategenerate an an octahedron octahedron of of 77 voxels voxels ; ; by by addingadding : : -- the the 12 12 secondsecond neighbours, neighbours, one one findsfinds a a cuboctahedron cuboctahedron of of 1919 voxelsvoxels (dist. (dist. ratio:ratio: 1.41)1.41) -- and and again again thethe 8 8 third third ones,ones, one one finallyfinally obtains obtains the the cube cube of of 27 27 voxelsvoxels (dist. (dist. ratio:ratio: 1.73).1.73). Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 5 CCCC Grid Grid NeighboursNeighbours Patterns Patterns •• The The 8 8 first first neighbours neighbours generategenerate a a cube cube of of 7 7 voxels; voxels; •• by by adding adding the the 6 6 second second neighbours,neighbours, one one findsfinds a a rhombododecahedronrhombododecahedron of of 1919 voxelsvoxels (distance (distance ratioratio 1.15);1.15); •• these these rhombododecahedrarhombododecahedra partitionpartition the the space space ; ; •• The The mediansmedians of of the the rhombsrhombs are are notnot edges.edges. Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 6 FCCFCC Grid Grid NeighboursNeighbours Patterns Patterns •• The The 12 12 first first neighboursneighbours generategenerate a a cube-octahedron cube-octahedron of of 1313 voxels. voxels. •• The The cube-octahedra cube-octahedra do do not not fillfill thethe spacespace (they (they leaveleave octahedricoctahedric holesholes betweenbetween them)them) •• However, However, they they generategenerate a a regularregular net net where where allall edgesedges havehave the the samesame lenght.lenght. Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 7 CCCC andand FCCFCC Grids Grids : : Square Square SectionsSections •• In In both both cc-grid cc-grid and and cfc-grid, cfc-grid, the the sections sections normalnormal to to –– the the cube cube diagonals diagonals are are composed composed of of hexagons, hexagons, –– the the cube cube edges edges are are composed composed of of squares.squares. •• The The ratios ratios between between square square andand interplane interplane spacingsspacings have have aa valuevalue 22 forfor cc-grid, cc-grid, and and 1.41 1.41 for for fcc fcc grid grid 11 1 Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 8 CCCC andand FCCFCC Grids Grids : : HexagonalHexagonal SectionsSections •• For For the the sections sections normalnormal to to thethe diagonal diagonal ofof the the cube, cube, we we findfind againagain aa staggering staggering effect, effect, with with now now a a perodicity perodicity of of three. three. •• The The ratios ratios between between hexagonal hexagonal and and interplane interplane spacings spacings are are 0.400.40 forfor cc-gridcc-grid and and 0.81 0.81 forfor fcc fcc gridgrid Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 9 SteinerSteiner DecompositionsDecompositions •• Consider Consider the the centre centre x x of of a a cube cube C C . By By dilating dilating –– the the three three vectors vectors from from x x to to the the faces faces of of C, C, one one obtains obtains a a CubeCube;; –– the the four four vectors vectors fromfrom x x to to thethe verticesvertices of of C,C, one one generatesgenerates a a RhombododecahedronRhombododecahedron;; –– the the six six vectors vectors fromfrom x x to to thethe edgesedges of of C,C, one one obtainsobtains a a TetrekaidecahedronTetrekaidecahedron;; –– the the two two tetrehedra tetrehedra built built on on the the diagonals diagonals of of C, C, one one obtains obtains a a Cube-octahedron.Cube-octahedron. Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 10 SteinerSteiner DecompositionsDecompositions (cont (contdd)) •• Cube Cube :: 1 1 1 (1,0) CC == (( ))⊕⊕ (( 1 )) ⊕⊕ (( )) •• Cube-octahedron Cube-octahedron : : 1 0 0 1 DD == (( 0 1 ))⊕⊕ (( 1 0 )) •• Rhombododecahedron Rhombododecahedron : : 1 . 0 . -1 . 0 RR == (( . 0 ))⊕⊕ (( . 1 )) ⊕⊕ (( 0 . ))⊕⊕ (( -1 . )) Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 11 SteinerSteiner DecompositionsDecompositions (cont (contdd)) By dilation of the six basic vectors of the f.c.c. grid , one obtains the tetrakaidecahedron ( and not the cube-octahedron, which is not Steiner). 0 1 0 -1 0 . 0 . 0 0 . ( . .)⊕( . )⊕( 1 .)⊕( -1 .)⊕( 0 . )⊕( . 0 ) Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 12 StaggeringStaggering thethe GridsGrids •• For For both both polyhedrapolyhedra R R andand D,D, the the previousprevious decompositionsdecompositions generategenerate thethe staggeringstaggering byby a a 45°45° turn turn of of the the grid,grid,e.g.e.g. 1 . 1 . 1 . 1 . D = 1 . 1 ∪ . 1 . ∪ 1 . 1 D =() (). 1 . ∪ ()()1 . 1 ∪ ()(). 1 . •• By By so so doing, doing, we we leave leave inside inside holes, holes, or, or, equivalently,we equivalently,we drop drop half half of of thethe voxels.voxels. Alernatively, Alernatively, we we cancan buildbuild more more condensed condensed R R andand DD asas followsfollows 1 1 . 1 . upper and lower planes 1 1 . odd central plane 1 1 1 upper and lower planes()(). odd central plane ()(). 1 . 1 . 1 1 upper and lower planes even central plane 1 1 1 upper and lower planes()(). 1 1 even central plane ()(). 1 . Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 13 DirectionsDirections •• Unlike Unlike the the unit unit disc, disc, the the unit unit sphere sphere cannot cannot be be divided divided into into as as manymany sectorssectors as as we we want.want. •• The The 3-D 3-D sphere sphere can can be be partitionned partitionned only only into into 2, 2, 4, 4, 6, 6, 8, 8, 12, 12, and and 20 20 equalequalsolidsolid angles. angles. •• The The axes axes ofof the the 6, 6, 88 andand 1212 solid solid angles angles coincide coincide withwith thethe directionsdirections ofof the the firstfirst neighboursneighbours in in the the cubic,cubic, cccc andand fcc fcc gridsgrids respectively.respectively. Therefore, Therefore, they they cancan bebe accessedaccessed digitally digitally.. •• The The finest finest equidistributed equidistributed digital digital set set of of directions directions is is written written : : 0 1 0 -1 0 . 0 . 0 0 . (( . )) ;; (( . )) ;; (( 1 . )) ;; ((-1 . )) ;; (( 0 . )) ;; (( . 0 )) Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 14 Cube-octahedralCube-octahedral DilationDilation a ) : Series of 74 cross sections of cube-octahedral dilation expanded Polystyrene ( from Shell) of size 5 of a ) Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 15 Cube-octahedralCube-octahedral closingclosing a ) Initial image cube-octahedral closing of size 12 of a ) Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 16 CubicCubic DilationDilation a ) cubic dilation of size 5 of a ) Ecole des Mines de Paris IWVF3, Capri, May 28-30, 1997 J Serra 17 SpaceSpace GraphsGraphs •In•In RR33 , , aa space space graph graph is is –a–a setset X Xofof pointspoints;; –– a a collectioncollection E Eofofedgesedges,, i.e.i.e.ofof lines lines homotopic homotopic to to the the segment segment [0,1][0,1] .. Both Both ends ends ofof each each edgeedge belongbelong totoXX,, andand two two edgesedges maymay possiblypossibly meet meet at at their their extremities extremities only only ; ; –– a a collectioncollection F Fofof faces faces,,i.e.i.e.ofof surfaces surfaces homotopic homotopic to to the the closed closed unitunit disc. disc. The The contours contours of of the the faces faces are are exclusively exclusively edges, edges, and and twotwo faces faces may may meetmeet alongalong edgesedges onlyonly ; ; –a–a setset P Pofof blocks blocks,, formed formed by by the the connected connected components components of of the the spacespace that that remain remain when when all all points points edges edges and and faces faces have have been been removed.removed.
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