**dodecahedron**- Its dual polyhedron is the rhombic dodecahedron. (wikipedia.org)
- The icosidodecahedron contains 12 pentagons of the dodecahedron and 20 triangles of the icosahedron: The icosidodecahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (3.n)2, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. (wikipedia.org)
- The pentakis snub dodecahedron is a convex polyhedron with 140 triangular faces, 210 edges, and 72 vertices. (wikipedia.org)
- In geometry, the excavated dodecahedron is a star polyhedron having 60 equilateral triangular faces. (wikipedia.org)
- It has the same external form as a certain facetting of the dodecahedron having 20 self-intersecting hexagons as faces. (wikipedia.org)
- A true excavated dodecahedron has the three congruent equilateral triangles as true faces of the polyhedron, while the interior equilateral triangle is not present. (wikipedia.org)
- It is the dual of the dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex. (wikipedia.org)
- Icosidodecahedron Pentakis dodecahedron is a slightly smaller Catalan solid which has 60 isosceles triangle faces, 90 edges (2 types), and 32 vertices (2 types). (wikipedia.org)
- Having twelve faces, it is a type of dodecahedron, although that name is usually associated with the regular polyhedral form with pentagonal faces. (wikipedia.org)
- In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron or dodecadeltahedron is a three-dimensional convex polyhedron with twelve equilateral triangles as its faces. (wikipedia.org)
- It is a dodecahedron, one of the eight deltahedra (convex polyhedra with equilateral triangle faces) and one of the 92 Johnson solids (non-uniform convex polyhedra with regular faces). (wikipedia.org)
- For example the icosahedron is {3,5+}1,0, and pentakis dodecahedron, {3,5+}1,1 is seen as a regular dodecahedron with pentagonal faces divided into 5 triangles. (wikipedia.org)
- Net In geometry, the chamfered cube (also called truncated rhombic dodecahedron) is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 (order 4) vertices. (wikipedia.org)
- This polyhedron looks similar to the uniform truncated octahedron: In geometry, the chamfered octahedron is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 8 (order 3) vertices. (wikipedia.org)
- If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is obtained (with pairs of coplanar triangles combined into rhombic faces). (wikipedia.org)
- The regular dodecahedron is the most common polyhedron in this class, being a platonic solid, with 12 congruent pentagonal faces. (wikipedia.org)

**edges**- The tetrakis cuboctahedron is a convex polyhedron with 32 triangular faces, 48 edges, and 18 vertices. (wikipedia.org)
- This polyhedron can be confused with a slightly smaller Catalan solid, the tetrakis hexahedron, which has only 24 triangles, 32 edges, and 14 vertices. (wikipedia.org)
- Octahedron with edges bisected and faces divided into subtriangles of the tetrakis cuboctahedron Cuboctahedron Tetrakis hexahedron The nonconvex octahemioctahedron looks like a concave tetrakis cuboctahedron with inverted square pyramids meeting at the polyhedron center. (wikipedia.org)
- In geometry, a regular icosahedron (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. (wikipedia.org)
- The pentakis icosidodecahedron or subdivided icosahedron is a convex polyhedron with 80 triangular faces, 120 edges, and 42 vertices. (wikipedia.org)
- It has 120 isosceles triangle faces (2 types), 180 edges (3 types) and 62 vertices (3 types). (wikipedia.org)
- In geometry, an edge-contracted icosahedron is a polyhedron with 18 triangular faces, 27 edges, and 11 vertices with C2v symmetry, order 4. (wikipedia.org)
- It can be constructed from the regular icosahedron, with one edge contraction, removing one vertex, 3 edges, and 2 faces. (wikipedia.org)
- It has 10 vertices, 23 edges, and 11 equilateral triangular faces and 2 trapezoid faces. (wikipedia.org)
- In geometry, a polyhedron is a solid in three dimensions with flat faces and straight edges. (wikipedia.org)
- Every edge has exactly two faces, and every vertex is surrounded by alternating faces and edges. (wikipedia.org)
- The smallest polyhedron is the tetrahedron with 4 triangular faces, 6 edges, and 4 vertices. (wikipedia.org)
- The resulting solid has 12 triangular faces, 8 vertices and 18 edges. (wikipedia.org)
- This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any n ≥ 7. (wikipedia.org)
- It follows from Euler's formula that any simplicial 2-sphere with n vertices has 3n − 6 edges and 2n − 4 faces. (wikipedia.org)
- A second type of geodesic passes near the intersection of the snub disphenoid with the plane that perpendicularly bisects the symmetry axis (the equator of the polyhedron), crossing the edges of eight triangles at angles that alternate between π/2 and π/6. (wikipedia.org)
- For three-dimensional simplicial polyhedra the numbers of edges and two-dimensional faces are determined from the number of vertices by Euler's formula, regardless of whether the polyhedron is stacked, but this is not true in higher dimensions. (wikipedia.org)
- Class II (b=c): {3,q+}b,b are easier to see from the dual polyhedron {q,3} with q-gonal faces first divided into triangles with a central point, and then all edges are divided into b sub-edges. (wikipedia.org)
- A geodesic polyhedron has straight edges and flat faces that approximate a sphere, but it can also be made as a spherical polyhedron (A tessellation on a sphere) with true geodesic curved edges on the surface of a sphere. (wikipedia.org)
- A polyhedron with e edges will have a chamfered form containing 2e new vertices, 3e new edges, and e new hexagonal faces. (wikipedia.org)
- The chamfer operation applied in series creates progressively larger polyhedra with new hexagonal faces replacing edges from the previous one. (wikipedia.org)
- The chamfered tetrahedron (or alternate truncated cube) is a convex polyhedron constructed as an alternately truncated cube or chamfer operation on a tetrahedron, replacing its 6 edges with hexagons. (wikipedia.org)
- It has 6 faces, 12 edges, and 8 vertices. (wikipedia.org)
- Only those terrain faces currently appearing within the pyramid of vision defined by the pilots eye and the edges of the pilots window need be displayed at any given time. (patents.com)
- These edges must be ~7% longer than the other edges, so the yellow triangles, unlike the other faces, are not quite regular - merely close. (robertlovespi.net)

**base and triangular faces**- The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. (yatzer.com)
- A solid figure with a polygonal base and triangular faces that meet at a common point. (thefreedictionary.com)

**Archimedean**- As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. (wikipedia.org)
- As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron. (wikipedia.org)
- Cundy & Rollett 1961 for archimedean polyhedra. (wikipedia.org)
- Chamfered Tetrahedron Chamfered Solids Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra Livio Zefiro VRML polyhedral generator (Conway polyhedron notation) VRML model Chamfered cube 3.2.7. (wikipedia.org)

**symmetry**- With D3d symmetry, order 12, it is a triangular gyrobicupola. (wikipedia.org)
- the canonical polyhedron chosen in this way has maximal symmetry among all choices of the canonical polyhedron. (wikipedia.org)
- Snub disphenoid Another 12-sided polyhedron with 2-fold symmetry and only triangular faces. (wikipedia.org)
- It exists first in a series of polyhedra with axial symmetry, so also can be given the name diagonal gyrobianticupola. (wikipedia.org)
- Because all its faces have an even number of sides with 180° rotation symmetry, it is a zonohedron. (wikipedia.org)
- The chamfered cube can be constructed with pyritohedral symmetry and rectangular faces. (wikipedia.org)
- The cube has three classes of symmetry, which can be represented by vertex-transitive coloring the faces. (wikipedia.org)
- The highest octahedral symmetry Oh has all the faces the same color. (wikipedia.org)
- It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry). (wikipedia.org)

**Octahedron**- It can also be topologically constructed from the octahedron, dividing each triangular face into 4 triangles by adding mid-edge vertices (an ortho operation). (wikipedia.org)
- The five octahedra defining any given icosahedron form a regular polyhedral compound, while the two icosahedra that can be defined in this way from any given octahedron form a uniform polyhedron compound. (wikipedia.org)
- The boundary of a convex polyhedron in R3 with triangular faces, such as an octahedron or icosahedron, is a simplicial 2-sphere. (wikipedia.org)
- The other three polyhedra with this property are the regular octahedron, the pentagonal bipyramid, and an irregular polyhedron with 12 vertices and 20 triangular faces. (wikipedia.org)

**snub**- The snub disphenoid name comes from Norman Johnson's 1966 classification of the Johnson solids, convex polyhedra all of whose faces are regular. (wikipedia.org)

**twenty triangular**- In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. (wikipedia.org)

**polyhedral**- One stronger form of the circle packing theorem, on representing planar graphs by systems of tangent circles, states that every polyhedral graph can be represented by a polyhedron with a midsphere. (wikipedia.org)

**geometry**- In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. (wikipedia.org)
- In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. (wikipedia.org)
- In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. (wikipedia.org)
- In geometry, the chamfered icosahedron is a convex polyhedron constructed from the rhombic triacontahedron by truncating the 20 order-3 vertices. (wikipedia.org)
- In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. (wikipedia.org)

**sides**- Egyptian pyramids are square in plan and their triangular sides, which directly face the points of the compass, slope upwards at approximately a 50° angle from the ground and meet at an apex. (thefreedictionary.com)
- 4. a solid having a polygonal base, and triangular sides that meet in a point. (thefreedictionary.com)
- A pyramid has TRIANGLE faces or sides meeting at an apex. (stormfront.org)
- None of them have TRIANGLE faces or sides meeting at an apex, indeed, none of your dirt mounds have an apex but rather a flattened top. (stormfront.org)

**Conway**- In Conway polyhedron notation it is represented by the letter c. (wikipedia.org)
- Conway polyhedron notation Near-miss Johnson solid Gallery of Wooden Polyhedra "Wooden polyhedra(English edition)" Goldberg, Michael (1937). (wikipedia.org)
- Truncated n-gonal trapezohedron - 2n pentagons, 2 n-gons, dual gyroelongated dipyramids Diminished trapezohedron Conway Notation for Polyhedra Try: "tndAn", where n=4,5,6. (wikipedia.org)

**deltahedra**- Generally they are described as (i) deltahedra, polyhedra with triangular faces or (ii) those same deltahedra with one or more vertices missing. (wikipedia.org)
- In three dimensions, every stacked polytope is a polyhedron with triangular faces, and several of the deltahedra (polyhedra with equilateral triangle faces) are stacked polytopes In a stacked polytope, each newly added simplex is only allowed to touch one of the facets of the previous ones. (wikipedia.org)

**twelve**- A regular icosahedron, therefore, includes twenty equilateral triangular faces and twelve vertices that are formed where points of five triangular faces meet. (google.com)
- Accordingly, a regular, truncated icosahedron is a polyhedron having thirty-two faces, twelve of which are equilateral pentagons and twenty of which are equilateral hexagons, and sixty vertices formed where the points of three faces meet. (google.com)

**convex polyhedra**- Convex Polyhedra with Regular Faces. (wikipedia.org)
- Consider all possible convex polyhedra which have regular polygons as faces. (robertlovespi.net)

**tilings**- With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors. (wikipedia.org)

**corresponding Goldberg polyhedron**- It is the dual of a corresponding Goldberg polyhedron, with mostly hexagonal faces. (wikipedia.org)

**meeting at each vertex**- It has five equilateral triangular faces meeting at each vertex. (wikipedia.org)

**equilateral triangle faces**- With all equilateral triangle faces, it has 2 sets of 3 coplanar equilateral triangles (each forming a half-hexagon), and thus is not a Johnson solid. (wikipedia.org)

**tetrahedron**- a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. (yatzer.com)
- However, the similar-looking pentagonal bipyramid is not a stacked polytope, because if it is formed by gluing tetrahedra together, the last tetrahedron will be glued to two triangular faces of previous tetrahedra instead of only one. (wikipedia.org)
- It can look a little like a truncated tetrahedron, , which has 4 hexagonal and 4 triangular faces, which is the related Goldberg polyhedron: GIII(1,1). (wikipedia.org)

**platonic**- Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have six faces, and because its faces cannot be equilateral triangles. (wikipedia.org)

**deltahedron**- According to these convensions, diborane would be the simplest nido- borane, as the 2-boron framework can be considered as the simplest deltahedron, the triangular dihedron or equilateral triangle, with one vertex removed. (wikipedia.org)

**common vertex**- 65. The implant of claim 61, wherein at least one of the one or more planar trusses comprises a plurality of triangular truss units having a common vertex. (freepatentsonline.com)

**pentagonal face**- The icosidodecahedron has four special orthogonal projections, centered on a vertex, an edge, a triangular face, and a pentagonal face. (wikipedia.org)

**geodesic**- It is a (2,1) geodesic polyhedron, made of all triangles. (wikipedia.org)
- Shifting a geodesic on the surface of the polyhedron by a small amount (small enough that the shift does not cause it to cross any vertices) preserves the property of being a geodesic and preserves its length, so both of these examples have shifted versions of the same type that are less symmetrically placed. (wikipedia.org)
- A geodesic polyhedron is a convex polyhedron made from triangles that approximates a sphere. (wikipedia.org)
- In Magnus Wenninger's Spherical models, polyhedra are given geodesic notation in the form {3,q+}b,c, where {3,q} is the Schläfli symbol for the regular polyhedron with triangular faces, and q-valance vertices. (wikipedia.org)
- The frequency of a geodesic polyhedron is defined by the sum of ν = b + c. (wikipedia.org)
- Two different geodesic polyhedra may have the same number of elements, for instance, {3,5+}5,3 and {3,5+}7,0 both have T=49. (wikipedia.org)
- Geodesic polyhedra are constructed by subdividing faces of simpler polyhedra, and then projecting the new vertices onto the surface of a sphere. (wikipedia.org)
- Geodesic polyhedra are the dual of Goldberg polyhedra. (wikipedia.org)
- Goldberg polyhedra are also related in that applying a kis operator (dividing faces triangles with a center point) creates new geodesic polyhedra, and truncating vertices of a geodesic polyhedron creates a new Goldberg polyhedron. (wikipedia.org)

**pyramids**- Tripentakis icosidodecahedron, the Kleetope of the icosahedron, can be obtained by raising low pyramids on each equilateral triangular face on a pentakis icosidodecahedron. (wikipedia.org)
- The nonconvex small icosihemidodecahedron looks like a pentakis icosidodecahedron with inverted pentagonal pyramids meeting at the polyhedron center. (wikipedia.org)
- A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. (wikipedia.org)
- An n-gonal truncated trapezohedron is a polyhedron formed by a n-gonal trapezohedron with n-gonal pyramids truncated from its two polar axis vertices. (wikipedia.org)

**triangles will be equilateral**- From this construction, all 80 triangles will be equilateral, but faces will be coplanar. (wikipedia.org)

**uniform**- Eight uniform star polyhedra share the same vertex arrangement. (wikipedia.org)
- The uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres. (wikipedia.org)
- The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123. (wikipedia.org)

**face**- The midsphere is so-called because, for polyhedra that have a midsphere, an inscribed sphere (which is tangent to every face of a polyhedron) and a circumscribed sphere (which touches every vertex), the midsphere is in the middle, between the other two spheres. (wikipedia.org)
- If O is the midsphere of a polyhedron P, then the intersection of O with any face of P is a circle. (wikipedia.org)
- The face planes of the polar polyhedron pass through the circles on O that are tangent to cones having the vertices of P as their apexes. (wikipedia.org)
- Any two polyhedra with the same face lattice and the same midsphere can be transformed into each other by a projective transformation of three-dimensional space that leaves the midsphere in the same position. (wikipedia.org)
- These are paths on the surface of the polyhedron that avoid the vertices and locally look like a shortest path: they follow straight line segments across each face of the polyhedron that they intersect, and when they cross an edge of the polyhedron they make complementary angles on the two incident faces to the edge. (wikipedia.org)
- For instance, the graphs of three-dimensional stacked polyhedra are exactly the Apollonian networks, the graphs formed from a triangle by repeatedly subdividing a triangular face of the graph into three smaller triangles. (wikipedia.org)
- For polyhedra, this operation adds a new hexagonal face in place of each original edge. (wikipedia.org)

**simplicial**- The most important open problem in the field is the g-conjecture, formulated by Peter McMullen, which asks about possible numbers of faces of different dimensions of a simplicial sphere. (wikipedia.org)
- The upper bound theorem gives upper bounds for the numbers fi of i-faces of any simplicial d-sphere with f0 = n vertices. (wikipedia.org)
- In other words, what are the possible sequences of numbers of faces of each dimension for a simplicial d-sphere? (wikipedia.org)
- There are other simplicial dodecahedra, such as the hexagonal bipyramid, but this is the only one that can be realized with equilateral faces. (wikipedia.org)
- It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. (wikipedia.org)
- One reason for the significance of stacked polytopes is that, among all d-dimensional simplicial polytopes with a given number of vertices, the stacked polytopes have the fewest possible higher-dimensional faces. (wikipedia.org)
- Analogously, the simplicial polytopes that maximize the number of higher-dimensional faces for their number of vertices are the cyclic polytopes. (wikipedia.org)

**triangle**- The subgrids can be extracted by looking at a triangular tiling, positioning a large triangle on top of grid vertices and walking paths from one vertex b steps in one direction, and a turn, either clockwise or counterclockwise, and then another c steps to the next primary vertex. (wikipedia.org)
- and spherical triangle faces. (wikipedia.org)
- Fold the triangle so that all of the vertices touch at a single point to form a triangular pyramid. (hmns.org)
- The term regular, when applied to an icosahedron, denotes a configuration wherein each of the twenty faces is an equally-dimensioned, equilateral triangle. (google.com)

**coplanar**- The dissected regular icosahedron is a name for this polytope with the two sets of 3 coplanar faces as trapezoids. (wikipedia.org)
- To join it, a polyhedron must fit the criteria for "Johnsonhood," except that some faces may be formed by amalgamation of multiple, coplanar regular polygons. (robertlovespi.net)

**pyramid**- As the orientation of the pyramid of vision changes in response to flight data, the displayed faces are correspondingly displaced, eventually moving out of the pyramid of vision. (patents.com)
- Faces which are currently not visible (outside the pyramid of vision) are clipped from the data flow. (patents.com)
- In addition, faces which are only partially outside of pyramid of vision are reconstructed to eliminate the outside portion. (patents.com)
- The outcodes (O.C.) are systematically processed and examined to determine which faces are completely inside the pyramid of vision (Case A--all signs positive), which faces are completely outside (Case C--All signs negative) and which faces must be reconstructed (Case B--both positive and negative signs). (patents.com)
- This creation is a truncated triangular pyramid . (hmns.org)

**hexagon**- The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron. (wikipedia.org)
- One of the star hexagon faces highlighted. (wikipedia.org)

**rhombic faces**- The original 12 rhombic faces become flattened hexagons, and the truncated vertices become squares. (wikipedia.org)

**topologically**- With six six-sided faces around each vertex, it is topologically equivalent to a quotient space of the hyperbolic order-6 hexagonal tiling, {6,6} and is an abstract type {6,6}6. (wikipedia.org)
- Notes: Polyhedra with different names that are topologically identical are listed together. (wikipedia.org)

**spheres**- This restrictive definition disallows the triangular bipyramid (as forming two tetrahedral holes rather than a single hole), pentagonal bipyramid (because the spheres for its apexes interpenetrate, so it cannot occur in sphere packings), and icosahedron (because it has interior room for another sphere). (wikipedia.org)

**Johnson**- If these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. (wikipedia.org)
- When Norman Johnson systematically found all of these, and named them, in the late 1960s, he found a number of other polyhedra which were extremely close to being in this set. (robertlovespi.net)
- Traditional near-misses involve relaxation of the rules for Johnson Solids to permit polyhedra with not-quite-regular faces to join a new "club. (robertlovespi.net)

**cuboctahedron**- end{aligned}}} The cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. (wikipedia.org)
- Its name comes from a topological construction from the cuboctahedron with the kis operator applied to the square faces. (wikipedia.org)

**vertices have four faces**- It is not a regular polyhedron because some vertices have four faces and others have five. (wikipedia.org)

**hexagonal trapezohedron**- hexagonal trapezohedron A similar 12-sided polyhedron with a twist and kite faces. (wikipedia.org)

**rectangular**- A massive monument of ancient Egypt having a rectangular base and four triangular faces culminating in a single apex, built over or around a crypt or tomb. (thefreedictionary.com)

**spherical**- Buckminster Fuller used these 6 great circles, along with 15 and 10 others in two other polyhedra to define his 31 great circles of the spherical icosahedron. (wikipedia.org)
- Magnus Wenninger's book Spherical Models explores these subdivisions in building polyhedron models. (wikipedia.org)

**correspond**- The horizon circles of a canonical polyhedron can be transformed, by stereographic projection, into a collection of circles in the Euclidean plane that do not cross each other and are tangent to each other exactly when the vertices they correspond to are adjacent. (wikipedia.org)
- In mathematical terms, the panels that form the casing of the traditional soccer ball correspond to the various faces of a regular, truncated icosahedron. (google.com)