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 P 4 2 P 4 2 22 P 4 2 2 1 2 n 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 2 d ( 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 2 d ( 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 2 d ( 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 = O 1 ∪ O 2 ∪ · · · ∪ O a , with p ∈ O 1 . 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 = O 1 ∪ O 2 ∪ · · · ∪ O a , with p ∈ O 1 . Let p ′ ∈ Gp \ p . Then, a necessary condition for Vor Gp ( p ) and Vor Gp ( 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 = O 1 ∪ O 2 ∪ · · · ∪ O a , with p ∈ O 1 . Let p ′ ∈ Gp \ p . Then, a necessary condition for Vor Gp ( p ) and Vor Gp ( p ′ ) to have a common facet is: If p ′ ∈ O 1 , that p ′ lies in (the relative interior of) a facet of 2 Vor O 1 ( 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 = O 1 ∪ O 2 ∪ · · · ∪ O a , with p ∈ O 1 . Let p ′ ∈ Gp \ p . Then, a necessary condition for Vor Gp ( p ) and Vor Gp ( p ′ ) to have a common facet is: If p ′ ∈ O 1 , that p ′ lies in (the relative interior of) a facet of 2 Vor O 1 ( p ). If p ′ �∈ O 1 , that p ′ lies in the interior of 2 Vor O 1 ( 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 = O 1 ∪ O 2 ∪ · · · ∪ O a , with p ∈ O 1 . Let p ′ ∈ Gp \ p . Then, a necessary condition for Vor Gp ( p ) and Vor Gp ( p ′ ) to have a common facet is: If p ′ ∈ O 1 , that p ′ lies in (the relative interior of) a facet of 2 Vor O 1 ( p ). If p ′ �∈ O 1 , that p ′ lies in the interior of 2 Vor O 1 ( p ). A volume argument implies that the first happens for at most 2(2 d − 1)times and the second at most 2 d times. Adding up gives a 2 d + 2 d − 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 Vor Gp ( 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 Vor Gp ( 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 Vor Gp ( 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 Vor Gp ( 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 Vor Gp ( 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 v’ 7 v’’ 3 1 3 5 4 7 1 3 6 8 2 2 8 2 7 8 1 v (b) (c) (a) P 4 2 2 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 1/4 1/2 4 2 2 I 4 1 2 2 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 = R 2 × 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 = R 2 × I ) R is the region between two parallel planes (call them horizontal). 2-h 1 P h 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 = R 2 × I ) R is the region between two parallel planes (call them horizontal). 2-h 1 P h 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 = R 2 × I ) R is the region between two parallel planes (call them horizontal). 2-h 1 P h 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 = R 2 × I ) R is the region between two parallel planes (call them horizontal). 2-h 1 P h 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 = R 2 × I ) R is the region between two parallel planes (call them horizontal). 2-h 1 P h 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 = R 2 × I ) R is the region between two parallel planes (call them horizontal). 2-h 1 P h 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 = R 2 × I ) Theorem (Bochis-S., 2001) Dirichlet stereohedra for groups with all reflections parallel have at most 15 facets. The bound is attained. P 2 2 2 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 “ P has at most seven neighbors h 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 “ P has at most seven neighbors h 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 “ P has at most seven neighbors h R at the other height” . -h -1 This follows from: h-2 Vor GP ( P ) and Vor GP ( P ′ ) with P ∈ { z = h } and P ′ ∈ { z = h ′ } can share a facet only if Vor GP ∩{ z = h } ( P ) and Vor GP ∩{ 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 “ P has at most seven neighbors h R at the other height” . -h -1 This follows from: h-2 Vor GP ( P ) and Vor GP ( P ′ ) with P ∈ { z = h } and P ′ ∈ { z = h ′ } can share a facet only if Vor GP ∩{ z = h } ( P ) and Vor GP ∩{ z = h ′ } ( P ′ ) overlap. Vor GP ∩{ z = h } and Vor GP ∩{ z = h ′ } are (infinite prisms over) planar Dirichlet tesselations for a certain group G h (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 “ P has at most seven neighbors h R at the other height” . -h -1 This follows from: h-2 Vor GP ( P ) and Vor GP ( P ′ ) with P ∈ { z = h } and P ′ ∈ { z = h ′ } can share a facet only if Vor GP ∩{ z = h } ( P ) and Vor GP ∩{ z = h ′ } ( P ′ ) overlap. Vor GP ∩{ z = h } and Vor GP ∩{ z = h ′ } are (infinite prisms over) planar Dirichlet tesselations for a certain group G h (the group go horizontal motions in G ). A planar Dirichlet region Vor G h p ( G h P ) can intersect at most seven regions of a Dirichlet tessellation Vor G h ( G h p ′ ) 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 G 1 × G 2 , where G i is a crystallographic group of dimension i ). We think of G 2 as horizontal and G 1 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 G 1 × G 2 , where G i is a crystallographic group of dimension i ). We think of G 2 as horizontal and G 1 as vertical. If “many” aspects are at the same height (that is, if the horizontal subgroup G 0 := G ∩ (1 × G 2 ) 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 G 1 × G 2 , where G i is a crystallographic group of dimension i ). We think of G 2 as horizontal and G 1 as vertical. If “many” aspects are at the same height (that is, if the horizontal subgroup G 0 := G ∩ (1 × G 2 ) 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 G 1 × G 2 , where G i is a crystallographic group of dimension i ). We think of G 2 as horizontal and G 1 as vertical. If “many” aspects are at the same height (that is, if the horizontal subgroup G 0 := G ∩ (1 × G 2 ) 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 G 0 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 G 1 × G 2 , where G i is a crystallographic group of dimension i ). We think of G 2 as horizontal and G 1 as vertical. If “many” aspects are at the same height (that is, if the horizontal subgroup G 0 := G ∩ (1 × G 2 ) 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 G 0 we can use: 4 for p 1, p 3, p 4 and p 6, 7 for p 2, 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 1/2 Non-cubic groups 1/2 1/2 Some examples 1/2 1/6 4/6 1/2 4/6 1/6 4/6 1/6 1/6 4/6 1/2 1/2 4/6 5/6 3/6 1/2 1/2 5/6 1/6 1/6 3/6 4/6 1/2 1/2 5/6 3/6 1/2 1/2 3/6 5/6 1/2 1/2 3/6 5/6 1/2 1/2 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 1/2 Non-cubic groups 1/2 1/2 Some examples 1/2 1/6 4/6 1/2 4/6 1/6 4/6 1/6 1/6 4/6 1/2 1/2 4/6 5/6 3/6 1/2 1/2 5/6 1/6 1/6 3/6 4/6 1/2 1/2 5/6 3/6 1/2 1/2 3/6 5/6 1/2 1/2 3/6 5/6 1/2 1/2 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 1/2 Non-cubic groups 1/2 1/2 Some examples 1/2 1/6 4/6 1/2 4/6 1/6 4/6 1/6 1/6 4/6 1/2 1/2 4/6 5/6 3/6 1/2 1/2 5/6 1/6 1/6 3/6 4/6 1/2 1/2 5/6 3/6 1/2 1/2 3/6 5/6 1/2 1/2 3/6 5/6 1/2 1/2 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 G 0 is a p 3, 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 1/2 Non-cubic groups 1/2 1/2 Some examples 1/2 1/6 4/6 1/2 4/6 1/6 4/6 1/6 1/6 4/6 1/2 1/2 4/6 5/6 3/6 1/2 1/2 5/6 1/6 1/6 3/6 4/6 1/2 1/2 5/6 3/6 1/2 1/2 3/6 5/6 1/2 1/2 3/6 5/6 1/2 1/2 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 G 0 is a p 3, 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 1/2 Non-cubic groups 1/2 1/2 Some examples 1/2 1/6 4/6 1/2 4/6 1/6 4/6 1/6 1/6 4/6 1/2 1/2 4/6 5/6 3/6 1/2 1/2 5/6 1/6 1/6 3/6 4/6 1/2 1/2 5/6 3/6 1/2 1/2 3/6 5/6 1/2 1/2 3/6 5/6 1/2 1/2 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 G 0 is a p 3, 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 1/2 Non-cubic groups 1/2 1/2 Some examples 1/2 1/6 4/6 1/2 4/6 1/6 4/6 1/6 1/6 4/6 1/2 1/2 4/6 5/6 3/6 1/2 1/2 5/6 1/6 1/6 3/6 4/6 1/2 1/2 5/6 3/6 1/2 1/2 3/6 5/6 1/2 1/2 3/6 5/6 1/2 1/2 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 G 0 is a p 3, 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 1/2 Non-cubic groups 1/2 1/2 Some examples 1/2 1/6 4/6 1/2 4/6 1/6 4/6 1/6 1/6 4/6 1/2 1/2 4/6 5/6 3/6 1/2 1/2 5/6 1/6 1/6 3/6 4/6 1/2 1/2 5/6 3/6 1/2 1/2 3/6 5/6 1/2 1/2 3/6 5/6 1/2 1/2 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 G 0 is a p 3, 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 1/2 Non-cubic groups 1/2 1/2 Some examples 1/2 1/6 4/6 1/2 4/6 1/6 4/6 1/6 1/6 4/6 1/2 1/2 4/6 5/6 3/6 1/2 1/2 5/6 1/6 1/6 3/6 4/6 1/2 1/2 5/6 3/6 1/2 1/2 3/6 5/6 1/2 1/2 3/6 5/6 1/2 1/2 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 G 0 is a p 3, 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 1/2 Non-cubic groups 1/2 1/2 Some examples 1/2 1/6 4/6 1/2 4/6 1/6 4/6 1/6 1/6 4/6 1/2 1/2 4/6 5/6 3/6 1/2 1/2 5/6 1/6 1/6 3/6 4/6 1/2 1/2 5/6 3/6 1/2 1/2 3/6 5/6 1/2 1/2 3/6 5/6 1/2 1/2 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 G 0 is a p 3, 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 G 0 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 G 0 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 G 0 be its horizontal subgroup , and consider the following parameters: a and a 0 , the numbers of aspects of G and G 0 . 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 G 0 be its horizontal subgroup , and consider the following parameters: a and a 0 , the numbers of aspects of G and G 0 . 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 G 0 be its horizontal subgroup , and consider the following parameters: a and a 0 , the numbers of aspects of G and G 0 . 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 G 0 : 4 for p 1, p 3, p 4 and p 6, and 7 for p 2, pg and pgg . F. Santos Number of facets of 3-d Dirichlet stereohedra
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