Visualization of polyhedra with Augmented Reality (AR) and Virtual Reality (VR) in A-frame
author: Paulo Henrique Siqueira - Universidade Federal do Paraná
contact: paulohscwb@gmail.com
versão em português
Pyramids, tetrahedrons, prisms and antiprisms
Augmented Reality | 3D Models | Home
Immersive rooms
Augmented Reality
To view polyhedra in AR, simply visit:
https://paulohscwb.github.io/polyhedra/polyhedron/ra.html
with any browser with a webcam device (smartphone, tablet or notebook).
Access to 3D models pages is done by clicking on the blue circle that appears on top of the marker.
3D models
1. Prism
U76 A general prism is a polyhedron possessing two congruent polygonal faces and with all remaining faces parallelograms. A right prism is a prism in which the top and bottom polygons lie on top of each other so that the vertical polygons connecting their sides are not only parallelograms, but rectangles. The regular right prisms have particularly simple nets, given by two oppositely-oriented n-gonal bases connected by a set of n squares. The example shown on this page is a regular right heptagonal prism.
Faces: 2 polygons of n sides (bases) and n squares, rectangles or parallelograms (side faces) | Edges: 3n | Vertices: 2n. More…
2. Stellated Prism
U78 A stellated or polygrammic prism is formed by two regular stellated polygons (polygrams) displaced along their axis of symmetry and with corresponding edges connected by lateral faces (squares, rectangles or parallelograms). The example shown on this page is of an octagonal stellated right prism (octagrammic prism).
Faces: 2 stellated polygons of n sides (bases) and n squares, rectangles or parallelograms (side faces) | Edges: 3n | Vertices: 2n. More…
3. Antiprism
U77 A general n-gonal antiprism is a polyhedron consisting of identical top and bottom n-gonal faces whose periphery is bounded by a set of 2n triangles with alternating up-down orientations. If the top and bottom faces are regular n-gons displaced relative to one another in the direction perpendicular to the plane of the polygons and rotated relative to one another by an angle of 180°/n, then the antiprism is known as a right antiprism and its faces are equilateral triangles. The example shown on this page is a regular hexagonal antiprism.
Faces: 2 polygons of n sides (bases) and 2n triangles (side faces) | Edges: 4n | Vertices: 2n. More…
4. Stellated Antiprism
U79 A stellated or polygrammic antiprism is formed by two upper and lower regular stellated polygons (polygrams), whose periphery is bounded by a set of 2n triangles with alternating orientations from top to bottom. The example shown on this page is of a pentagonal stellated right antiprism (pentagrammic antiprism).
Faces: 2 stellated polygons of n sides (bases) and 2n triangles (side faces) | Edges: 4n | Vertices: 2n. More…
5. Stellated Crossed Antiprism
U80 A stellated or polygrammic crossed antiprism is formed by two upper and lower regular stellated polygons (polygrams), whose periphery is bounded by a 2n set with alternating orientations from top to bottom connected with opposite vertices of the bases. The example shown on this page is a heptagonal stellated crossed right antiprism (heptagrammic crossed antiprism).
Faces: 2 stellated polygons of n sides (bases) and 2n triangles (side faces) | Edges: 4n | Vertices: 2n. More…
6. Pyramid
A pyramid is a polyhedron with one polygonal face (known as the “base”) and all the other faces triangles meeting at a common polygon vertex (known as the “apex”). A right pyramid is a pyramid for which the line joining the centroid of the base and the apex is perpendicular to the base. A regular pyramid is a right pyramid whose base is a regular polygon. The example shown on this page is of a regular right heptagonal pyramid.
Faces: 1 polygon of n sides (base) and n triangles (side faces) | Edges: 2n | Vertices: n + 1. More…
7. Stellated Pyramid
A stellated or polygrammic pyramid is formed by a regular stellated polygon (polygram) with corresponding edges connected by triangular side faces that meet at a common vertex (known as the “apex”). The example shown on this page is of an octagonal stellated pyramid (octagrammic pyramid).
Faces: 1 stellated polygon of n sides (base) and n triangles (side faces) | Edges: 2n | Vertices: n + 1. More…
8. Dipyramid
A dipyramid, also called a bipyramid or double pyramid, consists of two pyramids symmetrically placed base-to-base. The dipyramids are duals of the regular prisms. Their skeletons are the dipyramidal graphs. The example shown on this page is of a regular pentagonal dipyramid.
Faces: 2n triangles | Edges: 3n | Vertices: n + 2. More…
9. Stellated Dipyramid
A stellated dipyramid, also called a stellated bipyramid or stellated double pyramid, consists of two stellated pyramids symmetrically placed base-to-base. The stellated dipyramids are duals of the stellated prisms. The example shown on this page is of a regular pentagonal stellated dipyramid (pentagrammic dipyramid).
Faces: 2n triangles | Edges: 3n | Vertices: n + 2. More…
10. Trapezohedron
An n-trapezohedron, also called an antidipyramid, antibipyramid, or deltohedron is a solid composed of interleaved symmetric quadrilateral kites, half of which meet in a top vertex and half in a bottom vertex. A regular n-trapezohedron can be constructed from two sets of points placed around two regular n-gons displaced relative to one another in the direction perpendicular to the plane of the polygons and rotated relative to one another by an angle of 180°/n. This polyhedron is the dual of the antiprism. The example shown on this page is of a regular hexagonal trapezohedron.
Faces: 2n kites | Edges: 4n | Vertices: 2n + 2. More…
11. Stellated Trapezohedron
An stellated n-trapezohedron, also called an stellated antidipyramid, stellated antibipyramid, or stellated deltohedron is a solid composed of interleaved quadrilateral kites, half of which meet in a top vertex and half in a bottom vertex. This polyhedron is the dual of the stellated antiprism. The example shown on this page is of a regular heptagonal stellated trapezohedron (heptagrammic trapezohedron).
Faces: 2n kites | Edges: 4n | Vertices: 2n + 2. More…
12. Stellated Concave Trapezohedron
An stellated concave n-trapezohedron, also called an stellated concave antidipyramid, stellated concave antibipyramid, or stellated concave deltohedron is a solid composed of interleaved quadrilateral darts, half of which meet in a top vertex and half in a bottom vertex. This polyhedron is the dual of the stellated crossed antiprism. The example shown on this page is of a regular octagonal stellated concave trapezohedron (octagrammic concave trapezohedron).
Faces: 2n darts | Edges: 4n | Vertices: 2n + 2. More…
13. Isosceles tetrahedron
An isosceles tetrahedron is nonregular and each pair of opposite polyhedron edges are equal, so that all triangular faces are congruent. Isosceles tetrahedra are therefore isohedra. The only way for all the faces of a general tetrahedron to have the same perimeter or to have the same area is for them to be fully congruent, in which case the tetrahedron is isosceles. A tetrahedron is isosceles iff the sum of the face angles at each polyhedron vertex is 180°, and iff its insphere and circumsphere are concentric.
Faces: 4 scalene triangles | Edges: 6 | Vertices: 4. More…
14. Trapezo-rhombic Dodecahedron
The trapezo-rhombic dodecahedron, also called the rhombo-trapezoidal dodecahedron, is a general dodecahedron consisting of six identical rhombi and six identical isosceles trapezoids. The trapezo-rhombic dodecahedron can be obtained from the rhombic dodecahedron by slicing in half and rotating the two halves 60° with respect to each other. The lengths of the short and long edges of the rotated dodecahedron have lengths 2/3 and 4/3 times the length of the rhombic faces.
Faces: 6 rhombi and 6 isosceles trapezoids | Edges: 24 | Vertices: 14. More…
15. Octahedral Pentagonal Dodecahedron
The octahedral pentagonal dodecahedron, also called the pyritohedron, is made from 12 irregular pentagons with 4 equal sides and bilateral symmetry. The regular dodecahedron is a special case of this polyhedron.
Faces: 12 irregular pentagons | Edges: 30 | Vertices: 20. More…
16. Concave Dyakis Dodecahedron
The concave dyakis dodecahedron, also called the concave didodecahedron or concave diploid, is made from 24 quadrilaterals with only two equal and adjacent sides, with some dihedral angles greater than 180°. The Möbius octakis hexahedron is a special case of this polyhedron.
Faces: 24 quadrilaterals | Edges: 48 | Vertices: 26. More…
17. Dyakis Dodecahedron
The dyakis dodecahedron, also called the didodecahedron or diploid, is made from 24 quadrilaterals with only two equal and adjacent sides, with all dihedral angles less than 180°. The Möbius octakis hexahedron is a special case of this polyhedron.
Faces: 24 quadrilaterals | Edges: 48 | Vertices: 26. More…
18. Tetragonal Pentagonal Dodecahedron
A tetragonal pentagonal dodecahedron (also called of tetartoid, pentagon-tritetrahedron, or tetrahedric pentagon dodecahedron) is a dodecahedron with chiral tetrahedral symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes (each face have two pairs of equal adjacent sides). Although regular dodecahedra do not exist in crystals, the tetartoid form does.
Faces: 12 irregular pentagons | Edges: 30 | Vertices: 20. More…
19. Hexakis Tetrahedron
A hexakis tetrahedron (also called of hextetrahedron) is made by changing the length of the faces axes and edges-midpoint axes of a tetrahedron. The equilateral hexakis tetrahedron is a Möbius deltahedron.
Faces: 24 triangles | Edges: 36 | Vertices: 14. More…
20. Trapezohedral Tristetrahedron
The trapezohedral tristetrahedron is the polyhedra made from 12 kites shaped quadrilaterals. It can be constructed by means of symmetries from a regular tetrahedron.
Faces: 12 kites | Edges: 24 | Vertices: 14. More…
21. Iris Toroid
A polyhedron is called of toroid when had genus g ≥ 1 (i.e., one having one or more holes). A toroid is said to be non-regular if not all of its faces have the same number of vertices, or not all of its vertices join the same number of faces. The example shown is a regular heptagonal base, with square side faces.
Faces: n squares and 2n obtuse triangles | Edges: 5n | Vertices: 2n. More…
22. Iris Antitoroid
When we consider the triangular lateral faces on a toroid, we have a polyhedron called an antitoroid. The example shown is a regular heptagonal base, with equilateral triangles on the side faces.
Faces: 2n equilateral triangles and 2n obtuse triangles | Edges: 6n | Vertices: 2n. More…
23. Rhombic enneacontahedron
The rhombic enneacontahedron is a polyhedron composed of 90 rhombic faces with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim and is a zonohedron constructed from the 10 diameters of the dodecahedron with a superficial resemblance to the rhombic triacontahedron.
Faces: 90 rhombi | Edges: 180 | Vertices: 92. More…
24. Goddard-Henning enneahedron
The Goddard-Henning enneahedron is the canonical polyhedron obtained from the Goddard-Henning graph. It is a self-dual polyhedron and the bottom face is a square. The four faces sharing an edge with the bottom are isosceles triangles, and the remaining four faces that meet at the apex are “kites”.
Faces: 4 “kites”, 1 square and 4 isosceles triangles | Edges: 16 | Vertices: 9 | Dihedral angles: 101.53°, 120° and 104.51°. More…
25. Herschel enneahedron
The Herschel enneahedron is the canonical polyhedron whose skeleton is the Herschel graph. The dual polyhedron is a rectified triangular prism, which can be formed as the convex hull of the midpoints of the edges of a triangular prism.
Faces: 6 “kites” and 3 rhombi | Edges: 18 | Vertices: 9 | Dihedral angles: 107.01° and 119.11°. More…
26. Parallelepiped
The parallelepiped is a prism whose faces are all parallelograms. The term rhomboid is also sometimes used with meaning parallelepiped. The rectangular cuboid (six rectangular faces), cube (six square faces) and the rhombohedron (six rhombus faces) are all special cases of parallelepiped.
Faces: 6 parallelograms, squares, rectangles or rhombi | Edges: 12 | Vertices: 8. More…
27. Non convex Hexakis Tetrahedron
A non convex hexakis tetrahedron (also called of hextetrahedron) is made by changing the length of the faces axes and edges-midpoint axes of a tetrahedron. In this version, the corresponding vertices of the midpoints of the tetrahedron’s edges are closer to the center of the polyhedron.
Faces: 24 triangles | Edges: 36 | Vertices: 14. More…
28. Non convex Hexakis Tetrahedron v2
A non convex hexakis tetrahedron (also called of hextetrahedron) is made by changing the length of the faces axes and edges-midpoint axes of a tetrahedron. In this version, the corresponding vertices of the centers of the faces of the tetrahedron are closer to the center of the polyhedron.
Faces: 24 triangles | Edges: 36 | Vertices: 14. More…
29. Non convex Hexakis Tetrahedron v3
A non convex hexakis tetrahedron (also called of hextetrahedron) is made by changing the length of the faces axes and edges-midpoint axes of a tetrahedron. In this version, the corresponding vertices of the centers of the faces and the midpoints of the edges of the tetrahedron are closer to the center of the polyhedron.
Faces: 24 triangles | Edges: 36 | Vertices: 14. More…
30. Convex Trapezohedral Tristetrahedron
The convex trapezohedral tristetrahedron is the polyhedra made from 12 kites shaped quadrilaterals. It can be constructed by means of symmetries from a regular tetrahedron.
Faces: 12 kites | Edges: 24 | Vertices: 14. More…
Pyramids, tetrahedrons, prisms and antiprisms - Visualization of polyhedra with Augmented Reality and Virtual Reality by Paulo Henrique Siqueira is licensed with a license Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International.
How to cite this work:
Siqueira, P.H., "Pyramids, tetrahedrons, prisms and antiprisms - Visualization of polyhedra with Augmented Reality and Virtual Reality". Available in: <https://paulohscwb.github.io/polyhedra/polyhedron/>, March 2023.
References:
Weisstein, Eric W. “Archimedean Solid” From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/ArchimedeanSolid.html
Weisstein, Eric W. “Platonic Solid” From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/PlatonicSolid.html
Weisstein, Eric W. “Archimedean Dual” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/ArchimedeanDual.html
Weisstein, Eric W. “Uniform Polyhedron.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/UniformPolyhedron.html
Wikipedia https://en.wikipedia.org/wiki/Archimedean_solid
Wikipedia https://en.wikipedia.org/wiki/Platonic_solid
McCooey, David I. “Visual Polyhedra”. http://dmccooey.com/polyhedra/