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Polyhedron with 20 faces
has right dihedral angles. Rhombic icosahedron: a zonohedron made up of 20 congruent rhombs. It can be derived from the rhombic triacontahedron by removing
Icosahedron
Polyhedron with 20 rhombic faces
The rhombic icosahedron is a polyhedron shaped like an oblate sphere. Its 20 faces are congruent golden rhombi; 3, 4, or 5 faces meet at each vertex.
Rhombic_icosahedron
Catalan solid with 30 faces
icosidodecahedron. It is a zonohedron and can be seen as an elongated rhombic icosahedron. The ratio of the long diagonal to the short diagonal of each face
Rhombic_triacontahedron
Polyhedron with 12 faces
in non-periodic spacefillings along with the rhombic triacontahedron, the rhombic icosahedron, and rhombic hexahedra. There are 6,384,634 topologically
Dodecahedron
Polyhedron resembling a soccerball
In geometry, the truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. The polyhedron can
Truncated_icosahedron
Geometric operation which truncates the edges of polyhedra
chamfered icosahedron or tritruncated rhombic triacontahedron: by chamfering a regular icosahedron, or truncating the twenty order-3 vertices of the rhombic triacontahedron
Chamfer_(geometry)
Topics referred to by the same term
or the rhombic triacontahedron or the rhombic dodecahedral honeycomb or the rhombic icosahedron or the rhombic hexecontahedron or the rhombic enneacontahedron
Rhombic
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
Goldberg polyhedron with 42 faces
truncated icosahedron, which has 32 faces (12 pentagons and 20 hexagons) looks similar to the uniform chamfered dodecahedron (truncated rhombic triacontahedron)
Chamfered_dodecahedron
Convex polyhedron with 90 rhombic faces
In geometry, a rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 rhombic faces; with three, five, or six rhombi
Rhombic_enneacontahedron
Quadrilateral with sides of equal length
of 90 rhombic faces, with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim ones. The rhombic icosahedron is a polyhedron
Rhombus
5-dimensional hypercube
graphs. The 5-cube can be projected down to 3 dimensions with a rhombic icosahedron envelope. There are 22 exterior vertices, and 10 interior vertices
5-cube
Near-miss Johnson solid with 92 faces
truncated icosahedron, jtI, but it is topologically equivalent to the rhombic enneacontahedron with all rhombic faces. The rectified truncated icosahedron can
Rectified truncated icosahedron
Rectified_truncated_icosahedron
Any of the five regular polyhedra
the regular dodecahedron (twelve pentagonal faces), and the regular icosahedron (twenty triangular faces). Geometers have studied the Platonic solids
Platonic_solid
Generalisation of dice with identical faces
to be noble. Not all isozonohedra are isohedral. For example, a rhombic icosahedron is an isozonohedron but not an isohedron. A polyhedron (or polytope
Isohedral_figure
Metabidiminished icosahedron, Hexagonal bipyramid, Hexagonal trapezohedron, Triakis tetrahedron, Rhombic dodecahedron, Hendecagonal pyramid, Trapezo-rhombic dodecahedron
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Catalan solid with 120 faces
the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces
Disdyakis_triacontahedron
Polyhedron with 12 congruent golden rhombus faces
of ten and eight golden rhombic faces with parallel edges. Removing only one zone of ten faces produces the rhombic icosahedron. Removing three zones of
Bilinski_dodecahedron
Polyhedron with 30 faces
rhombs. What resembles an "excavated" rhombic triacontahedron (compare excavated dodecahedron and excavated icosahedron) can be seen within the middle of
Great_rhombic_triacontahedron
Jessen's icosahedron Near-miss Johnson solid Parallelepiped Pentagonal bifrustum Polytetrahedron Pyritohedron Rhombic enneacontahedron Rhombic icosahedron Rhombo-hexagonal
List_of_mathematical_shapes
Polyhedral compound
and an Icosahedron shares the same vertices as a list of other polyhedra, including the Rhombic triacontahedron and the Small triambic icosahedron. In the
Compound of dodecahedron and icosahedron
Compound_of_dodecahedron_and_icosahedron
regular icosahedron. A subset of these are illustrated in Wenninger (1989), alongside constructions for physical models (W19–W66). The rhombic dodecahedron
List of polyhedral stellations
List_of_polyhedral_stellations
Polyhedron with 30 faces
the dual of the medial rhombic triacontahedron, the dodecadodecahedron. Great rhombic triacontahedron Great triambic icosahedron Wenninger, Magnus (1983)
Medial rhombic triacontahedron
Medial_rhombic_triacontahedron
rhombi Rhombic dodecahedron - 12 rhombi Rhombic icosahedron - 20 rhombi Rhombic triacontahedron - 30 rhombi Rhombic hexecontahedron - 60 rhombi Rhombic enneacontahedron
Rhombic_hectotriadiohedron
34th model as his 9th stellation of the icosahedron Triakis icosahedron Small triambic icosahedron Medial rhombic triacontahedron The Regular Polyhedra
Great_triambic_icosahedron
Rhombus with diagonals in the golden ratio
rhombic dodecahedron (with 12 faces), the rhombic icosahedron (with 20 faces), the rhombic triacontahedron (with 30 faces), and the nonconvex rhombic
Golden_rhombus
Natural number between 89 and 91
golden ratio. The obtuse angle of the broad rhombic faces is also the dihedral angle of a regular icosahedron, with the obtuse angle in the faces of golden
90_(number)
Convex polyhedron projected from hypercube
congruent rhombic faces: Bilinski's rhombic dodecahedron. Rhombic icosahedron Rhombohedron There are infinitely many zonohedra with rhombic faces that
Zonohedron
Extending the elements of a polytope to form a new figure
stellation of the icosahedron, the great stellated dodecahedron (the final stellation of the dodecahedron) and the fourth stellation of the rhombic dodecahedron
Stellation
Number, approximately 1.618
its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle – that is, a rectangle with an aspect ratio of
Golden_ratio
Flat-sided three-dimensional shape
example, HIV is enclosed in a regular icosahedron, as is the head of a typical myovirus. The regular icosahedron may also appeared in the applications
Polyhedron
Archimedean solid with 62 faces
icosidodecahedral rhombus being his name for a rhombic triacontahedron. There are different truncations of a rhombic triacontahedron into a topological rhombicosidodecahedron:
Rhombicosidodecahedron
connected. Its dual polyhedron is a rhombic hemi-dodecahedron which has 7 vertices (1-7), 12 edges (a-l), and 6 rhombic faces (A-F). It has a real presentation
Hemi-cuboctahedron
Polyhedron with 32 faces
inscribed within and sharing the edges of the icosahedron. The dual of the great icosidodecahedron is the great rhombic triacontahedron; it is nonconvex, isohedral
Great_icosidodecahedron
Ordered chemical structure with no repeating pattern
A 6-cube projected into the rhombic triacontahedron using the golden ratio in the basis vectors. This is used to understand the aperiodic icosahedral
Quasicrystal
13 polyhedra; duals of the Archimedean solids
adjacent faces is the same. Additionally, two Catalan solids, the rhombic dodecahedron and rhombic triacontahedron, are edge-transitive, meaning their edges are
Catalan_solid
Polyhedron with 8 triangles and 6 squares
edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron. The cuboctahedron can be constructed in many ways: Its construction
Cuboctahedron
Two polyhedral compounds with the same name
There are two different compounds of great icosahedron and great stellated dodecahedron: one is a dual compound and a stellation of the great icosidodecahedron
Compound of great icosahedron and great stellated dodecahedron
Compound_of_great_icosahedron_and_great_stellated_dodecahedron
3D symmetry group
regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron
Icosahedral_symmetry
Solid with 12 equal pentagonal faces
has a relation with other Platonic solids, one of them is the regular icosahedron as its dual polyhedron. Other new polyhedra can be constructed by using
Regular_dodecahedron
Shape with ten sides
projection plane of the 5-cube. A dissection is based on 10 of 30 faces of the rhombic triacontahedron. The list OEIS: A006245 defines the number of solutions
Decagon
Polytope or tiling with one type of edge
isotoxal, but not isohedral. Their duals, including the rhombic dodecahedron and the rhombic triacontahedron, are isohedral and isotoxal, but not isogonal
Isotoxal_figure
Polyhedron with regular congruent polygons as faces
perhaps only one has been found to be a true icosahedron, as opposed to a reinterpretation of the icosahedron dual, the dodecahedron. It is also possible
Regular_polyhedron
Polyhedron with 24 faces
projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of geodesic domes. It has four Wythoff constructions
Dodecadodecahedron
Catalan solid with 60 faces
identical to the nonconvex rhombic hexecontahedron. The deltoidal hexecontahedron can be derived from a dodecahedron (or icosahedron) by pushing the face centers
Deltoidal_hexecontahedron
Archimedean solid with 32 faces
There is another way to construct it, and that is rectification of an icosahedron or a dodecahedron. Convenient Cartesian coordinates for the vertices
Icosidodecahedron
Polyhedron with two kinds of faces
dual-pair cube and octahedron, in the first case, and of the dual-pair icosahedron and dodecahedron, in the second case. These forms representing a pair
Quasiregular_polyhedron
( − 5 5 ) {\displaystyle \arccos(-{\frac {\sqrt {5}}{5}})} 116.565° Icosahedron {3,5} (3.3.3.3.3) arccos ( − 5 3 ) {\displaystyle \arccos(-{\frac {\sqrt
Table of polyhedron dihedral angles
Table_of_polyhedron_dihedral_angles
Honeycomb made from unique polyhedrons
honeycomb is a uniform honeycomb, constructed from dodecahedron, icosahedron, and icosidodecahedron cells, in a rhombicosidodecahedron vertex figure
Dodecahedral-icosahedral honeycomb
Dodecahedral-icosahedral_honeycomb
Polyhedral compound
dodecahedron and icosahedron Compound of small stellated dodecahedron and great dodecahedron Compound of great stellated dodecahedron and great icosahedron Wenninger
Compound of cube and octahedron
Compound_of_cube_and_octahedron
American speaker, author and toymaker
The Ball of Whacks consists of 30 rhombic pyramid shaped magnetic blocks. These pieces combine to form a rhombic triacontahedron. They can be combined
Roger_von_Oech
Four-dimensional analogue of the cube
has a rhombic dodecahedral envelope. Two vertices of the tesseract are projected to the origin. There are exactly two ways of dissecting a rhombic dodecahedron
Tesseract
Twelve-sided combination puzzle
Rhombic dodecahedron version of the puzzle with equivalent mechanism
Skewb_Ultimate
Construction set toy
stellations of the rhombic triacontahedron Many stellations of the regular icosahedron Zonohedra, especially the rhombic enneacontahedron and rhombic triacontahedron
Zometool
icosidodecahedron Great rhombic triacontahedron 2|5/23 (5/2.3)2 Ih U54 K59 30 60 32 20{3}+12{5/2} 95 Truncated great icosahedron Great stellapentakis dodecahedron
List of Wenninger polyhedron models
List_of_Wenninger_polyhedron_models
Convex polyhedron whose faces are almost regular polygons
alternated cubic honeycomb, ignoring any obscured faces. Examples: 3.3.3.3.3.3 Rhombic prism Wedge Trigonal trapezohedron Gyroelongated trigonal pyramid Triangulated
Near-miss_Johnson_solid
regular dodecahedron, and regular icosahedron. The regular octahedron is dual to the cube, and the regular icosahedron is dual to the regular dodecahedron
Dual_uniform_polyhedron
Polytope made by turning a polytope's facets into pyramids
octahedron is the Kleetope of an octahedron, and the triakis icosahedron is the Kleetope of an icosahedron. These Kleetopes are formed by adding a triangular pyramid
Kleetope
Polyhedron with 60 faces
kites Pentagonal hexecontahedron - pentagons Triakis icosahedron - isosceles triangles Concave Rhombic hexecontahedron - rhombi 27 uniform star-polyhedral
Hexecontahedron
as follows: The medial rhombic triacontahedron and dodecadodecahedron are dual to each other. The medial triambic icosahedron and ditrigonal dodecadodecahedron
List_of_regular_polytopes
Polyhedron related to sphere packing
Tetrahedron: W1 O3*, W2 O3*, W1 O3, W1 O4 Octahedron: W2 O1, W1 O6 Cube: W2 O6 Icosahedron and dodecahedron have no representation as Waterman polyhedra.[citation
Waterman_polyhedron
Polyhedron associated with another by swapping vertices for faces
canonical dual compound. Canonical dual compound of cuboctahedron (light) and rhombic dodecahedron (dark). Pairs of edges meet on their common midsphere. Dual
Dual_polyhedron
Method of describing higher-order polyhedra
dodecahedron mC = mO Pentagonal icositetrahedron gC = gO Rhombic triacontahedron jD = jI Triakis icosahedron kI = nD Pentakis dodecahedron kD = nI Deltoidal hexecontahedron
Conway_polyhedron_notation
Polyhedral compound
stellation of the icosahedron, and given as Wenninger model index 23. It can be constructed by a rhombic triacontahedron with rhombic-based pyramids added
Compound_of_five_octahedra
bipyramid Elongated square bipyramid Metabidiminished icosahedron Augmented tridiminished icosahedron Sphenocorona 11 Decagonal pyramid Nonagonal bipyramid
List of small polyhedra by vertex count
List_of_small_polyhedra_by_vertex_count
Multi-stage paper folding technique
can form any equilateral polyhedron including those having rhombic faces, like the rhombic dodecahedron. The Mukhopadhyay module can form any equilateral
Modular_origami
Marked objects for finding random numbers
zodiac signs, and the third represents the 12 houses. A specialized icosahedron die provides the answers of the Magic 8 Ball, conventionally used to
Dice
Polytope with highest degree of symmetry
3\}} , icosahedron { 3 , 5 } {\displaystyle \{3,5\}} , dodecahedron { 5 , 3 } {\displaystyle \{5,3\}} — and the four star polyhedra—great icosahedron { 3
Regular_polytope
3D shape made of polyhedra sharing a common center
(compound of icosahedron and great dodecahedron) and the great complex icosidodecahedron (compound of small stellated dodecahedron and great icosahedron). If
Polytope_compound
cuboctohedron. Four faces around the vertex in the pattern p.q.r.q. The name rhombic stems from inserting a square in the cuboctahedron and icosidodecahedron
List of uniform polyhedra by Wythoff symbol
List_of_uniform_polyhedra_by_Wythoff_symbol
Set of points described by relative position in space
self-intersecting great dodecahedron shares its edge arrangement with the convex icosahedron: A group polytopes that share both a vertex arrangement and an edge arrangement
Vertex_arrangement
Polyhedron with 9 faces
isospectral polyhedral graphs, enneahedra with eight vertices each. Slicing a rhombic dodecahedron in half through the long diagonals of four of its faces results
Enneahedron
Notation for a polyhedron's vertex figure
figure, with vertex configuration is (5.5.5.5.5)/2 or (55)/2. A great icosahedron, {3,5/2} also has a pentagrammic vertex figure, with vertex configuration
Vertex_configuration
Rhombic dodecahedron, a solid with hidden lines that thinner and a lighter shade of grey.
Hidden_line
Polygon rhombic triacontahedron (quasi-regular dual: face- and edge-uniform) (Video) icosidodecahedron 30 60 32 rhombus triakis icosahedron (Video) truncated
Solids with icosahedral symmetry
Solids_with_icosahedral_symmetry
Geometry book by Bonnie Stewart
from the golden rhombus, including the Bilinski dodecahedron, rhombic icosahedron, and rhombic triacontahedron. The second edition also includes the Császár
Adventures_Among_the_Toroids
Empirical study of systems in transformation
constant proportionality of the octahedron, the cube, the rhombic triacontahedron, and the rhombic dodecahedron when referenced to the tetrahedron as volumetric
Synergetics_(Fuller)
1948 wood engraving print by M. C. Escher
within the print also include all of the five Platonic solids and the rhombic dodecahedron. In order to depict polyhedra accurately, Escher made models
Stars_(M._C._Escher)
Pseudocylindrical equal-area map projection
H=6 HEALPix has similarities to another alternative grid based on the icosahedron. List of map projections Spatial grid Geodesic grid Calabretta, Mark
HEALPix
families, each with two regular forms, and one quasiregular form. All have rhombic duals of the quasiregular form, but only one is shown: Here's 3 example
List of isotoxal polyhedra and tilings
List_of_isotoxal_polyhedra_and_tilings
Shape in hyperbolic geometry
{\displaystyle 90^{\circ }=\pi /2=\pi (1-{\tfrac {2}{4}})} , and an ideal regular icosahedron, with five edges per vertex, has dihedral angles 108 ∘ = 3 π / 5 = π
Ideal_polyhedron
Measure of how closely a shape resembles a sphere
{\displaystyle {\frac {\sqrt[{3}]{36\pi +18\pi {\sqrt {2}}}}{6}}\approx 0.9633} Rhombic triacontahedron 4 5 + 2 5 s 3 {\displaystyle 4{\sqrt {5+2{\sqrt {5}}}}\
Sphericity
Geometric operation on a regular polytope
bipyramid Square antiprism or tetragonal trapezohedron Cuboctahedron or rhombic dodecahedron Icosidodecahedron or rhombic triacontahedron Image Animation
Cantellation_(geometry)
Allotrope of carbon
карбо-s-икосаэдре" [On hypothetical systems: carbon dodecahedron, S-icosahedron and carbon-S-icosahedron]. Dokl. Akad. Nauk SSSR. 209: 610. Iijima, S (1980). "Direct
Fullerene
Removal of alternate vertices
cube Rhombic dodecahedron Dual of cuboctahedron Truncated rhombic dodecahedron Rhombic triacontahedron Dual of icosidodecahedron Truncated rhombic triacontahedron
Alternation_(geometry)
Property of objects which appear unchanged after a partial rotation
15×2-fold axes: the rotation group I of order 60 of a dodecahedron and an icosahedron. The group is isomorphic to alternating group A5. The group contains
Rotational_symmetry
Polytope or tiling whose vertices are identical
This truncated rhombic dodecahedron is 2-isogonal because it contains two transitivity classes of vertices. This polyhedron is made of squares and flattened
Isogonal_figure
Edge-joined polygons which fold into a polyhedron
Prism Truncated Icosahedron Truncated dodecahedron Truncated cube Truncated octahedron Truncated cuboctahedron Rhombicuboctahedron Rhombic dodecahedron
Net_(polyhedron)
Polytope constructed from alternation of a hypercube
but can be stretched with different lengths in n-axes of symmetry. The rhombic disphenoid is the three-dimensional example as alternated cuboid. It has
Demihypercube
Four-dimensional analogues of the regular polyhedra in three dimensions
faces with large ones in same planes. (Example: an icosahedron greatens into a great icosahedron) aggrandizement – replaces the cells with large ones
Regular_4-polytope
Shapes for crystals with interfaces and twins
fiveling by Rose, while for three boundaries per unit, a close-to-perfect icosahedron is formed. (An image of an 0.5 cm gold mineral crystal is shown later
Extended_Wulff_constructions
Four-dimensional geometric object with flat sides
tetrahedral cells) 27 unique convex dual uniform honeycombs, including: Rhombic dodecahedral honeycomb Disphenoid tetrahedral honeycomb Others: Weaire–Phelan
4-polytope
crossed trapezoid vertex figures. The first column include the convex rhombic polyhedra, created by inserting two squares into the vertex figures of
List of uniform polyhedra by vertex figure
List_of_uniform_polyhedra_by_vertex_figure
Sculpture in Pennsylvania State University's mathematics department
cube (6 faces), octahedron (8 faces), dodecahedron (12 faces), and icosahedron (20 faces). They have been known since the time of the Ancient Greeks
Octacube_(sculpture)
Classification system for symmetry groups in geometry
and is also represented more compactly as [3[ ]×[ ]], representing the rhombic symmetry of the Coxeter diagram. The paracompact complete graph diagram
Coxeter_notation
3D combination puzzle
the octahedron (Skewb Diamond), the dodecahedron (Megaminx), and the icosahedron (Dogic). There are also puzzles that change shape such as Rubik's Snake
Rubik's_Cube
Regular object in four dimensional geometry
polytope boundary, an analogy of nearly the cuboctahedron and its dual the rhombic dodecahedron in four dimensions. It is composed of 24 octahedral cells
24-cell
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RHOMBIC ICOSAHEDRON
RHOMBIC ICOSAHEDRON
RHOMBIC ICOSAHEDRON
RHOMBIC ICOSAHEDRON
RHOMBIC ICOSAHEDRON
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RHOMBIC ICOSAHEDRON
RHOMBIC ICOSAHEDRON
RHOMBIC ICOSAHEDRON
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