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Polyhedron of the Day #271: Great truncated icosidodecahedron

The great truncated icosidodecahedron is a uniform star polyhedron. It has 62 faces (30 squares, 20 hexagons, 12 decagrams), 180 edges, and 120 vertices. It may also be referred to as the great quasitruncated icosidodecahedron or the stellatruncated icosidodecahedron. Its Bowers-style acronym is gaquatid. It can be constructed through quasicantitruncation of the great stellated dodecahedron or great icosahedron, or through quasitruncation of the great icosidodecahedron (after which edge lengths are made equal). It has icosahedral symmetry. Its vertex figure is a scalene triangle. Its dual is the great disdyakis triacontahedron.

Maeder, R.E. (1997). "Great truncated icosidodecahedron (68)." [Image]. MathConsult AG. https://www.mathconsult.ch/static/unipoly/68.html.

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Polyhedron of the Day #270: Bisquare icosahedron

The bisquare icosahedron is a near-miss Johnson solid. It has 20 faces (2 quadrilaterals, 18 scalene triangles), 31 edges, and 13 vertices. Its dual is not uniquely named.

Image and polyhedron model created using Robert Webb's Stella software, found at https://www.software3d.com/Stella.php.

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Polyhedron of the Day #269: Small icosihemidodecacron

The small icosihemidodecacron is a uniform dual star polyhedron. It has 30 faces, 60 edges, and 26 vertices. As its dual has faces passing through its centre, this polyhedron has corresponding vertices at infinity. This is represented in the images above by intersecting prisms extending an arbitrarily long distance in both directions. It is visually identical to the small dodecahemidodecacron. It has icosahedral symmetry. Its dual is the small icosihemidodecahedron.

Weisstein, Eric W. "Small icosihemidodecacron." [Image]. From MathWorld—A Wolfram Resource. https://mathworld.wolfram.com/SmallIcosihemidodecacron.html.

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Polyhedron of the Day #268: Parabigyrate rhombicosidodecahedron

The parabigyrate rhombicosidodecahedron is a Johnson solid (J73). It has 62 faces (20 triangles, 30 squares, 12 pentagons), 120 edges, and 60 vertices. Its Bowers-style acronym is pabgyrid. It can be constructed by rotating two opposing pentagonal cupolaic caps of the rhombicosidodecahedron by 36 degrees. It is canonical. Its dual is not uniquely named.

Left image created by AndrewKepert using Cyp's povray macros, CC BY-SA 3.0, https://upload.wikimedia.org/wikipedia/commons/4/48/Parabigyrate_rhombicosidodecahedron.png.

Right image created using Robert Webb's Stella software, found at https://www.software3d.com/Stella.php.

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Polyhedron of the Day #267: Compound of three tetrahedra

The compound of three tetrahedra is a prismatic uniform polyhedral compound. It has 12 faces (all triangles), 24 edges, and 12 vertices. It consists of three tetrahedra rotated by 60 degrees with respect to each other. It shares its vertex arrangement with the hexagonal antiprism. It has dihedral symmetry. It is self-dual.

Left image created using Robert Webb's Stella software, found at http://www.software3d.com/Stella.php.

Weisstein, Eric W. "Tetrahedron 3-compound." [Image]. From MathWorld—A Wolfram Resource. https://mathworld.wolfram.com/Tetrahedron3-Compound.html

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Polyhedron of the Day #266: Sesquitruncated hexagonal bifrustum

The sesquitruncated hexagonal bifrustum is a near-miss Johnson solid. It has 56 faces (36 isosceles triangles, 6 rectangles, 12 decagons, 2 dodecagons), 138 edges, and 84 vertices. It can be constructed by inscribing decagons in the trapezoidal faces and dodecagons in the hexagonal faces of the hexagonal bifrustum, then filling in the remaining gaps with isosceles triangles and rectangles. Its dual is not uniquely named.

Image created using Robert Webb's Stella software, found at https://www.software3d.com/Stella.php.

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Polyhedron of the Day #265: Elongated gyrobifastigium

The elongated gyrobifastigium is a convex polyhedron. It has 8 faces (4 rectangles, 4 pentagons), 18 edges, and 12 vertices, all edges having the same length. It is also known as the gabled rhombohedron or elongated digonal gyrobicupola. It can self-tessellate space. It can be constructed by inserting a cube between the two halves of the gyrobifastigium. Its dual is the snub disphenoid.

Weisstein, Eric W. "Elongated gyrobifastigium." [Image]. From MathWorld—A Wolfram Resource. https://mathworld.wolfram.com/ElongatedGyrobifastigium.html

Right image created by Tomruen, CC BY-SA 4.0, https://upload.wikimedia.org/wikipedia/commons/a/a5/Dual_digonal_gyrobianticupola.png.

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Polyhedron of the Day #264: Small retrosnub icosicosidodecahedron

The small retrosnub icosicosidodecahedron is a uniform star polyhedron. It has 112 faces (100 triangles, 12 pentagrams), 180 edges, and 60 vertices. It may also be called the retrosnub disicosidodecahedron, small inverted retrosnub icosicosidodecahedron, or retroholosnub icosahedron. Its Schläfli symbol is sr{5/3,3/2}. Its Bowers-style acronym is sirsid. It has icosahedral symmetry. Its vertex figure is a nonconvex mirror-symmetric hexagon. Its dual is the small hexagrammic hexecontahedron.

Maeder, R.E. (1997). "Small retrosnub icosicosidodecahedron (72)." [Image]. MathConsult AG. https://www.mathconsult.ch/static/unipoly/72.html.

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Polyhedron of the Day #263: Tri-tridiminished icosahedron

The tri-tridiminished icosahedron is a convex polyhedron. It has 9 faces (6 triangles, 3 kites), 15 edges, and 8 vertices. Its Bowers-style acronym is titdi. It can be constructed by removing 3 sets of 3 vertices from the icosahedron. Its dual is the tridiminished icosahedron, from which it derives its name.

Image created using Robert Webb's Stella software, found at http://www.software3d.com/Stella.php.

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Polyhedron of the Day #262: Ditetragonal prism

The ditetragonal prism is a semi-uniform polyhedron. It has 10 faces (8 rectangles, 2 ditetragons), 24 edges, and 16 vertices. Its Bowers-style acronym is detip. As the name suggests, it is a prism based on the ditetragon. Its vertex figure is a scalene triangle. Its dual is the tetrambic tegum.

Image created using Robert Webb's Stella software, found at https://www.software3d.com/Stella.php.

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Polyhedron of the Day #261: Augmented truncated dodecahedron

The augmented truncated dodecahedron is a Johnson solid (J68). It has 42 faces (25 triangles, 5 squares, 1 pentagon, 11 decagons), 105 edges, and 65 vertices. Its Bowers-style acronym is autid. It is constructed by attaching a pentagonal cupola to one of the decagonal faces of the truncated dodecahedron. Its dual is not uniquely named.

Left image created by AndrewKepert using Cyp's povray macros, CC BY-SA 3.0, https://upload.wikimedia.org/wikipedia/commons/d/dc/Augmented_truncated_dodecahedron.png.

Right image created using Robert Webb's Stella software, found at https://www.software3d.com/Stella.php.

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Polyhedron of the Day #260: Compound of five small cubicuboctahedra

The compound of five small cubicuboctahedra is a uniform polyhedral compound. It has 100 faces (40 triangles, 30 squares, 30 octagons), 240 edges, and 120 vertices. It is also referred to as the rhombihyperhombicosicosahedron. Its Bowers-style acronym is rahrie. It has the same vertex arrangement as the compound of 5 small rhombicuboctahedra. It has icosahedral symmetry. Its vertex figure is a crossed isosceles trapezoid. Its dual is the compound of five small hexacronic icositetrahedra.

Image created using Robert Webb's Stella software, found at http://www.software3d.com/Stella.php.

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Polyhedron of the Day #259: Pentakis icosidodecahedron

The pentakis icosidodecahedron is a convex polyhedron. It has 80 faces (all triangles), 120 edges, and 42 vertices. It can also be referred to as the subdivided icosahedron. It can be constructed by augmenting pyramids onto the pentagonal faces of the icosidodecahedron or by dividing each triangular face of the icosahedron into 4 by adding mid-edge vertices. Since this produces coplanar faces, edges are then lengthed to make each vertex the same distance from the polyhedron's centre. It has icosahedral symmetry. Its dual is the chamfered dodecahedron.

Right image taken as a screenshot from David I. McCooey's Visual Polyhedra website, https://dmccooey.com/polyhedra/PentakisIcosidodecahedron.html.

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Polyhedron of the Day #258: Great deltoidal hexecontahedron

The great deltoidal hexecontahedron is a uniform dual star polyhedron. It has 60 faces (all darts), 120 edges, and 62 vertices. It is also known as the great sagittal ditriacontahedron or the great strombic hexecontahedron. It is visually indistinguishable from the great rhombidodecacron, which has fewer vertices and different faces. Its dual is the nonconvex great rhombicosidodecahedron.

Left image created using Robert Webb's Stella software, found at http://www.software3d.com/Stella.php.

Weisstein, Eric W. "Great deltoidal hexecontahedron." [Image]. From MathWorld—A Wolfram Resource. https://mathworld.wolfram.com/GreatDeltoidalHexecontahedron.html.

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Polyhedron of the Day #257: Great icosihemidodecahedron

The great icosihemidodecahedron is a uniform star polyhedron. It has 26 faces (20 triangles, 6 decagrams), 60 edges, and 30 vertices. Its Bowers-style acronym is geihid. As is suggested by its name, it is a hemipolyhedron. It shares its edge arrangement with the great icosidodecahedron (which it is a faceting of) and the great dodecahemidodecahedron. It has icosahedral symmetry. Its vertex figure is a crossed quadrilateral. Its dual is the great icosihemidodecacron.

Maeder, R. E. (1997). "Great icosihemidodecahedron (71)." [Image]. MathConsult AG. https://www.mathconsult.ch/static/unipoly/71.html.

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Polyhedron of the Day #256: Hexagonal antiprism

The hexagonal antiprism is a prismatic uniform polyhedron. It has 14 faces (2 hexagons, 12 triangles), 24 edges, and 12 vertices. Its Schläfli symbol can be written as s{2,12} or sr{2,6}. Its Bowers-style acronym is hap. Attaching triangular cupolae to one or both of its bases produces Johnson solids (the gyroelongated triangular cupola and the gyroelongated triangular bicupola, respectively). Its vertex figure is an isosceles trapezoid. Its dual is the hexagonal trapezohedron.

Right image created using Robert Webb's Stella software, found at http://www.software3d.com/Stella.php.

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Polyhedron of the Day #255: Triakis triangular tiling

The triakis triangular tiling is a uniform dual tiling of the Euclidean plane. Its faces are isosceles triangles, with 3 or 12 meeting at a vertex. It may also be called the kisdeltille, as it may be constructed by applying a kis operation to the deltille (triangular tiling). Its dual is the truncated hexagonal tiling.

Left image created by Tomruen, CC BY-SA 4.0, https://upload.wikimedia.org/wikipedia/commons/5/54/1-uniform_4_dual.svg.

Right image created by T. Piesk, CC BY 4.0, https://upload.wikimedia.org/wikipedia/commons/2/2f/Tiling_truncated_6_dual_simple.svg.