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Physical manifestation of a geometric shape
A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less
Polyhedron_model
Isogonal polyhedron with regular faces
Wenninger polyhedron models Polyhedron Regular polyhedron Quasiregular polyhedron Semiregular polyhedron Polyhedron model Pseudo-uniform polyhedron Uniform
Uniform_polyhedron
polyhedra from the book Polyhedron Models, by Magnus Wenninger. The book was written as a guide book to building polyhedra as physical models. It includes templates
List of Wenninger polyhedron models
List_of_Wenninger_polyhedron_models
Flat-sided three-dimensional shape
In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many' and ἕδρον (-hedron) 'base, seat') is a three-dimensional figure
Polyhedron
Polyhedron made from triangles that approximates a sphere
A geodesic polyhedron is a convex polyhedron made from triangles which approximates a sphere. They usually have icosahedral symmetry, such that they have
Geodesic_polyhedron
Edge-joined polygons which fold into a polyhedron
of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra
Net_(polyhedron)
Outermost stellation of the icosahedron
the book. In Wenninger's book Polyhedron Models, the final stellation of the icosahedron is included as the 17th model of stellated icosahedra with index
Final stellation of the icosahedron
Final_stellation_of_the_icosahedron
Toroidal polyhedron with 7 faces
Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces. The tetrahedron and the Szilassi polyhedron are the only
Szilassi_polyhedron
Extending the elements of a polytope to form a new figure
stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form
Stellation
Uniform star polyhedron with 124 faces
great snub disicosidisdodecahedron) is a degenerate nonconvex uniform polyhedron, indexed last as U75. It has 124 faces (40 triangles, 60 squares, and
Great dirhombicosidodecahedron
Great_dirhombicosidodecahedron
Topics referred to by the same term
Physical models of mathematical objects, created for instructional or artistic purposes, including: Polyhedron model, a physical model of a polyhedron Mathematical
Mathematical model (disambiguation)
Mathematical_model_(disambiguation)
Non-convex polyhedron with no triangulation
In geometry, a Schönhardt polyhedron is a polyhedron with the same combinatorial structure as a regular octahedron, but with dihedral angles that are non-convex
Schönhardt_polyhedron
Kepler–Poinsot polyhedron
In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {5⁄2, 3}. It is one of four nonconvex regular polyhedra
Great_stellated_dodecahedron
Concave polyhedron
stellation of the icosahedron. It appears in Magnus Wenninger's book Polyhedron Models as model 28, the third stellation of icosahedron. All 20 vertices and 30
Excavated_dodecahedron
Convex polyhedron with regular faces
Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons and that is not a uniform polyhedron. There are 92 such solids: 48 composed
Johnson_solid
Polyhedron associated with another by swapping vertices for faces
In geometry, every polyhedron is associated with a second dual structure, wherein the vertices of one correspond to the faces of the other and the edges
Dual_polyhedron
American mathematician (1919–2017)
17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Born to German immigrants
Magnus_Wenninger
Solid with six equal square faces
these edges form six square faces of the same size. It is an example of a polyhedron. It is a special case of a cuboid, a parallelepiped, and a rhombohedron
Cube
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i
List_of_uniform_polyhedra
Toroidal polyhedron with 14 triangle faces
geometry, the Császár polyhedron (Hungarian: [ˈt͡ʃaːsaːr]) is a nonconvex toroidal polyhedron with 14 triangular faces. This polyhedron has no diagonals;
Császár_polyhedron
Any of the five regular polyhedra
Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical
Platonic_solid
Polyhedron with regular congruent polygons as faces
regular polyhedron is a polyhedron with regular and congruent polygons as faces. Its symmetry group acts transitively on its flags. A regular polyhedron is
Regular_polyhedron
Graph in which every two vertices are adjacent
edge set of a triangle, K4 a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K7 as
Complete_graph
Kepler–Poinsot polyhedron with 20 faces
"uniform polyhedra Great icosahedron". Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. Coxeter, Harold Scott
Great_icosahedron
Line segment joining two adjacent vertices in a polygon or polytope
a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment
Edge_(geometry)
Book on stellations of the regular icosahedron by H. S. M. Coxeter and colleagues
In this book a polyhedron is stellated by extending the face planes of a polyhedron until they meet again to form a new polyhedron or compound. When
The_Fifty-Nine_Icosahedra
dual uniform polyhedron is the dual of a uniform polyhedron. Where a uniform polyhedron is vertex-transitive, a dual uniform polyhedron is face-transitive
Dual_uniform_polyhedron
Planar maps require at most four colors
based on the Petersen graph, giving a 6-coloring. Interactive Szilassi polyhedron model. Each of the seven faces is adjacent to every other – in the SVG image
Four_color_theorem
Catalan solid with 24 kite faces
Weisstein, Eric W., "Deltoidal icositetrahedron" ("Catalan solid") at MathWorld. Deltoidal (Trapezoidal) Icositetrahedron – Interactive Polyhedron model
Deltoidal_icositetrahedron
Self-intersecting uniform polyhedron
polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can
Uniform_star_polyhedron
Polyhedron with 92 faces
dodecahedron and the great icosidodecahedron. In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great inverted snub icosidodecahedron
Great_snub_icosidodecahedron
Catalan solid with 60 faces
In geometry, a pentakis dodecahedron or kisdodecahedron is a polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron;
Pentakis_dodecahedron
Prism with a 5-sided base
all regular, the pentagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the third in an infinite set of prisms formed
Pentagonal_prism
Polyhedra in which all vertices are the same
removed] is an interactive 3D polyhedron viewer that allows you to save the model in SVG, STL, or OBJ format. Stella: Polyhedron Navigator: Software used to
Archimedean_solid
Shape in hyperbolic geometry
In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather
Ideal_polyhedron
Conic solid with a polygonal base
A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral
Pyramid_(geometry)
Polyhedron resulting from the snub operation
a snub polyhedron is a polyhedron obtained by performing a snub operation: alternating a corresponding omnitruncated or truncated polyhedron, depending
Snub_polyhedron
Kepler–Poinsot polyhedron
In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {5⁄2, 5}. It is one of four
Small_stellated_dodecahedron
Polyhedron resembling a soccerball
truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. The polyhedron can be regarded as a football
Truncated_icosahedron
Catalan solid with 24 faces
model Archived 2021-11-22 at the Wayback Machine Conway Notation for Polyhedra Try: "dtO" or "kC" Tetrakis Hexahedron – Interactive Polyhedron model The
Tetrakis_hexahedron
Catalan solid with 120 faces
triacontahedron has the most faces of any other strictly convex polyhedron where every face of the polyhedron has the same shape. Projected into a sphere, the edges
Disdyakis_triacontahedron
Polyhedron with 92 faces
uniform star polyhedron, indexed as U69. It is given a Schläfli symbol sr{5⁄3,3}, and Coxeter-Dynkin diagram . In the book Polyhedron Models by Magnus Wenninger
Great inverted snub icosidodecahedron
Great_inverted_snub_icosidodecahedron
Polyhedron with 7 faces
geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges,
Tetrahemihexahedron
Uniform star polyhedron with 204 faces
dirhombidodecahedron, also called Skilling's figure, is a degenerate uniform star polyhedron. Its full realization has 204 faces (120 triangles, 60 squares, and 24
Great disnub dirhombidodecahedron
Great_disnub_dirhombidodecahedron
Polyhedron with 10 faces
In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. It has 10 faces (6 squares and 4 regular hexagons), 24 edges and
Cubohemioctahedron
Catalan solid with 48 faces
Weisstein, Eric W., "Disdyakis dodecahedron" ("Catalan solid") at MathWorld. Disdyakis Dodecahedron (Hexakis Octahedron) Interactive Polyhedron Model
Disdyakis_dodecahedron
Method of describing higher-order polyhedra
Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified
Conway_polyhedron_notation
Catalan solid with 24 faces
Octahedron – Interactive Polyhedron Model Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra VRML model Archived 2018-10-11 at
Triakis_octahedron
Catalan solid with 60 faces
or a tetragonal hexacontahedron) is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, an Archimedean solid. It is one of six
Deltoidal_hexecontahedron
Polyhedral compound
compounds of hyperbolic tilings. This polyhedron is the first stellation of the icosidodecahedron, and given as Wenninger model index 47. The stellation facets
Compound of dodecahedron and icosahedron
Compound_of_dodecahedron_and_icosahedron
Shape with six sides
grid Central place theory Cube picture Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595, archived from
Hexagon
Solid with 12 equal pentagonal faces
regular dodecahedron or pentagonal dodecahedron is a dodecahedron (a polyhedron with 12 faces) composed of regular pentagonal faces, three meeting at
Regular_dodecahedron
Compound polyhedron
chiricosahedron, is one of the five regular polyhedral compounds. This compound polyhedron is also a stellation of the regular icosahedron. It was first described
Compound_of_five_tetrahedra
Any of 4 regular star polyhedra
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating and faceting the regular convex dodecahedron
Kepler–Poinsot_polyhedron
Antiprism with 6-sided caps
nonconvex polyhedron with 24 equilateral triangles. Weisstein, Eric W. "Antiprism". MathWorld. Hexagonal Antiprism: Interactive Polyhedron model Virtual
Hexagonal_antiprism
Antiprism with a five-sided base
Antiprism: Interactive Polyhedron Model Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra VRML model Archived 2021-08-14 at
Pentagonal_antiprism
different kinds of faces. List of Wenninger polyhedron models The fifty nine icosahedra M Wenninger, Polyhedron models; Cambridge University Press, 1st Edn (1983)
Stellation_diagram
Solid with twenty equal triangular faces
The regular icosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from a pentagonal antiprism by attaching two pentagonal
Regular_icosahedron
Archimedean solid with 62 faces
Placement of Regular Polygons Craig S. Kaplan Wenninger, Magnus (1974), Polyhedron Models, Cambridge University Press, ISBN 978-0-521-09859-5, MR 0467493 Cromwell
Truncated_icosidodecahedron
Pyramid with a square base
Square Pyramid – Interactive Polyhedron Model Virtual Reality Polyhedra georgehart.com: The Encyclopedia of Polyhedra (VRML model Archived 7 October 2023 at
Square_pyramid
Archimedean solid with 26 faces
2026-09-10. Wenninger, Magnus (1974), Polyhedron Models, Cambridge University Press, ISBN 978-0-521-09859-5, MR 0467493 (Model 15, p. 29) Williams, Robert (1979)
Truncated_cuboctahedron
Catalan solid with 60 faces
hexecontahedron ) Weisstein, Eric W., "Pentagonal hexecontahedron" ("Catalan solid") at MathWorld. Pentagonal Hexecontrahedron – Interactive Polyhedron Model
Pentagonal_hexecontahedron
Theorem on graph coloring on surfaces
Horizontally rotatable Szilassi polyhedron model with each of 7 faces adjacent to every other. In the SVG image, move the mouse to rotate it.
Heawood_conjecture
Polyhedron with 44 faces
dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons)
Small_dodecicosidodecahedron
Spatial grid based on a geodesic polyhedron
based on a geodesic polyhedron or Goldberg polyhedron. The earliest use of the (icosahedral) geodesic grid in geophysical modeling dates back to 1968 and
Geodesic_grid
Two tetrahedra crossing each other
octahedra). Wenninger, Magnus J. (1971), "The stellated octahedron (19)", Polyhedron Models, Cambridge University Press, p. 37, doi:10.1017/CBO9780511569746.010
Stellated_octahedron
Uniform star polyhedron with 12 faces
geometry, the octahemioctahedron or octatetrahedron is a nonconvex uniform polyhedron, indexed as U3. It contains twelve faces (eight triangles and four hexagons)
Octahemioctahedron
Polyhedron with eight triangular faces
In geometry, an octahedron (pl.: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid
Octahedron
Polyhedral compound
compounds of two tetrahedra"). This polyhedron is a stellation of the icosahedron, and given as Wenninger model index 25. It is also a facetting of the
Compound_of_ten_tetrahedra
Shape with ten sides
& Sons, p. 146, ISBN 9780471461630. Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595. The elements
Decagon
Nonconvex uniform polyhedron with 20 faces
Robert. "Great Cubicuboctahedron". Stella: Polyhedron Navigator. Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371
Great_cubicuboctahedron
In geometry, a monostatic polytope or unistable polyhedron is a d {\displaystyle d} -polytope which "can stand on only one face". They were described in
Monostatic_polytope
In geometry, the small triambic icosahedron is a star polyhedron composed of 20 intersecting non-regular hexagon faces. It has 60 edges and 32 vertices
Small_triambic_icosahedron
Uniform star polyhedron whose faces pass through its center
uniform star polyhedron some of whose faces pass through its center. These hemi-faces lie parallel to the faces of some other symmetrical polyhedron, and their
Hemipolyhedron
Polyhedron with 12 faces
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with
Dodecahedron
Catalan solid with 60 faces
Wenninger, Magnus (1974). "22: The great stellated dodecahedron". Polyhedron Models. Cambridge University Press. pp. 40–42. ISBN 0-521-09859-9. Zaks,
Triakis_icosahedron
Polygon shape with eight sides
Islamic architecture Smoothed octagon Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595. Dao Thanh Oai
Octagon
Polyhedron with 10 faces
infinite family of trapezohedra, face-transitive polyhedra. Its dual polyhedron is the pentagonal antiprism. As a decahedron it has ten faces which are
Pentagonal_trapezohedron
Polyhedral compound
The compound of five octahedra is one of the five regular polyhedron compounds, and can also be seen as a stellation. It was first described by Edmund
Compound_of_five_octahedra
Sphere tangent to every edge of a polyhedron
or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform
Midsphere
Sculpture conceived via mathematics
Mingei, 2021 ISBN 978-2-9566150-3-3 OCLC 1268517010 Wenninger, Magnus: Polyhedron Models. Cambridge: Cambridge University Press, 1971 ISBN 978-0-521-09859-5
Mathematical_sculpture
Number of windings of a polytope around its center of symmetry
In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions, representing
Density_(polytope)
Polyhedron with 62 faces
geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U67. It has 62 faces (20 triangles, 30 squares and 12 pentagrams)
Nonconvex great rhombicosidodecahedron
Nonconvex_great_rhombicosidodecahedron
Catalan solid with 12 faces
convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron
Rhombic_dodecahedron
Polyhedron with 32 faces
In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges
Great_icosidodecahedron
Archimedean solid with 8 faces
all of its vertices off, a process known as truncation. The resulting polyhedron has 4 equilateral triangles and 4 regular hexagons, 18 edges, and 12 vertices
Truncated_tetrahedron
3D shape made of polyhedra sharing a common center
form a convex polyhedron called its convex hull. A compound is a faceting of its convex hull.[citation needed] Another convex polyhedron is formed by the
Polytope_compound
Archimedean solid with 26 faces
The rhombicuboctahedron or small rhombicuboctahedron is a polyhedron with 26 faces, consisting of 8 equilateral triangles and 18 squares. It was named
Rhombicuboctahedron
Polyhedron with 6 rhombic and 6 trapezoidal faces
The trapezo-rhombic dodecahedron is a polyhedron with six trapezoidal and six rhombic faces. This polyhedron can be obtained from a rhombic dodecahedron
Trapezo-rhombic_dodecahedron
Geometric operation which truncates the edges of polyhedra
chamfer or edge-truncation is a topological operator that modifies one polyhedron into another. It separates the faces by reducing them, and adds a new
Chamfer_(geometry)
Uniform star polyhedron with 112 faces
icosicosidodecahedron, or retroholosnub icosahedron) is a nonconvex uniform polyhedron, indexed as U72. It has 112 faces (100 triangles and 12 pentagrams), 180
Small retrosnub icosicosidodecahedron
Small_retrosnub_icosicosidodecahedron
Polyhedron formed by joining two prisms
In geometry, the gyrobifastigium is a polyhedron that is constructed by attaching a triangular prism to the square face of another one. It is an example
Gyrobifastigium
Polyhedron with 84 faces
dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60. It is given a Schläfli symbol sr{5/3,5}. Let ξ ≈ 2.109759446579943
Inverted snub dodecadodecahedron
Inverted_snub_dodecadodecahedron
Polyhedron with 32 faces
"63: great dodecicosahedron". MathConsult. Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. pp. 9–10. Weisstein
Great_dodecicosahedron
Solid with 2 parallel n-gonal bases connected by n parallelograms
In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy (rigidly moved without rotation) of the
Prism_(geometry)
Sphere tangent to every face of a polyhedron
or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces. It is the largest
Inscribed_sphere
Polyhedron with 8 rhombic and 4 hexagonal faces
parallelohedron identified by Evgraf Fedorov. In other words, it is a space-filling polyhedron, meaning the elongated dodecahedron and its copy can tile space face-to-face
Elongated_dodecahedron
Catalan solid with 12 faces
triangular faces. The triakis tetrahedron is a Catalan solid, the dual polyhedron of a truncated tetrahedron, an Archimedean solid with four hexagonal and
Triakis_tetrahedron
Canadian geometer (1907–2003)
Diagrams". arXiv:2601.02290 [math.RT]. Wenninger, Magnus J. (1971). Polyhedron Models. Cambridge University Press. Goldberg, M. "Review of Regular Polytopes"
H._S._M._Coxeter
Polyhedral compound
great stellated dodecahedron and great icosahedron Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 978-0-521-09859-5. v t e
Compound of cube and octahedron
Compound_of_cube_and_octahedron
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