We will continue our discussion of Platonic solids here by moving into the topic of variations of the Platonic solids.  These include stellations, truncations, rotations, expansions and compounds.  These include, but are not limited to, the 13 Archimedean solids and their duals the Catalan solids.

# Uniform Polyhedra – 75 total

### 53 semi-regular non-convex polyhedra

A display of uniform polyhedra at the Science Museum in London.  For more info visit:  https://en.wikipedia.org/wiki/Uniform_star_polyhedron

# Compound Polyhedra

“By the duality principle, for every polyhedron there exists another polyhedron in which faces and polyhedron vertices occupy complementary locations.  This polyhedron is known as the dual, or reciprocal…The dual of a Platonic solid or Archimedeans solid can be computed by connecting the midpoints of the sides surrounding each polyhedron vertex.”1

Recall from Article 41 that the duals are as follows:

• Tetrahedron – Inverted Tetrahedron

• Octahedron – Cube

• Icosahedron – Dodecahedron

Compound polyhedra are formed from the five Platonic solids and their duals.  The compound also represents the mid-point or point of balance between the transition from one dual to another.

These compounds include:

## Star Tetrahedron (stellated octahedron) – 1440°

It is composed of 1 tetrahedron pointing up & 1 pointing down.

If you place eight tetrahedra on each of the 8 faces of an octahedron you get the stellated octahedron, or star tetrahedron seen here.

## 1. Cuboctahedron (vector equilibrium) – cube and octahedron combined = 3600°

It has: 8 triangular faces; 6 square faces; 14 total faces; 24 edges; 12 vertices

This is the first Archimedean solid we look at.  It has square and equilateral triangle faces.

Net of the cuboctahedron:

## 2. Icosidodecahedron – icosahedron and dodecahedron combined = 10080°

It has: 20 triangular faces; 12 pentagonal faces; 32 total faces; 60 edges; 30 vertices

This is second Archimedean solid.  It has pentagonal and triangular faces.

The net of the icosidodecahedron:

# The Archimedean Solids

There are 13 Archimedean solids in number.  They are named after Archimedes (287-212 BC) who discussed them in a now-lost work.

The Archimedean solids are semi-regular convex polyhedra.  They have very high symmetry.

They are composed of regular polygons.  (Like Platonic solids)

They have regular faces of more than 1 type.  (Unlike Platonic Solids)

They have identical vertices. (Like Platonic Solids)

They all fit perfectly within a sphere with tetrahedral, octahedral or icosahedral symmetry. (Like Platonic Solids)

Each Archimedean solid is formed from a Platonic solid.  This occurs through truncation (5), expansion (4), compounds (2) and expansion plus rotation (2).  Each of these will be outlined below.

Rotation of cuboctahedral array of 12 Archimedean polyhedra

The number of vertices is 720º divided by the vertex angle defect.

Each has 1 circumsphere and 1 midsphere (intersphere).

Each has an insphere for each type of face.

The larger faces have smaller inspheres touching their centers.

The circumsphere surrounds the entire solid.  The points of the solid will touch on the circumsphere without projecting through them.

The midsphere or intersphere runs through the center points of each edge.

The insphere fits entirely inside the solid resting against the midpoints of each face.

Nets of the Archimedean solids are shown below:

 Volume Face Type # of Faces # of Edges # of Corners Sum of Degrees Icosidodecahedron 20 triangles; 12 pentagons 32 60 30 10080 Great rhombicuboctahedron 12 squares; 8 hexagons; 6 octagons 26 72 48 16560 Great rhombiicosidodecahedron 30 squares; 20 hexagons; 12 decagons 62 180 120 42480 Cuboctahedron 8 triangles; 6 squares 14 24 12 3600 Rhombicuboctahedron 8 triangles; 18 squares 26 48 24 7920 Snub Cube 32 triangles; 6 squares 38 60 24 7920 Rhombicosadodecahedron 20 triangles; 30 squares; 12 pentagons 62 120 60 20880 Snub Dodecahedron 80 triangles; 12 pentagons 92 150 60 20880 Truncated Tetrahedron 4 triangles; 4 hexagons 8 18 12 3600 Truncated Octahedron 6 squares; 8 hexagons 14 36 24 7920 Truncated Cube 8 triangles; 6 octagons 14 36 24 7920 Truncated Icosahedron 12 pentagons; 20 hexagons 32 90 60 20880 Truncated Dodecahedron 20 triangles; 12 decagons 32 90 60 20880 Sum of All 452 984 558 165,600

# Five Truncations of the Platonic Solids

To truncate means to cut off the corners so they are replaced by surfaces.  The surfaces will then have equal edge lengths.

This is called uniform truncation.

“Truncation of a three-valent vertex will generate a triangle; four-valent vertices become squares, and so on. The number of edges determines the number of sides of the new polygon.”2 Amy Edmondson

Each truncated solid defines 4 concentric spheres due to the larger faces having the smaller inspheres touching their centers.

Each can sit neatly inside their original Platonic Solid & its Dual.

We saw the first two Archimedean solids above.  Now we will look at the five truncations.

All five truncations of the Platonic solids are Archimedean solids.

These are:

## 3. Truncated tetrahedron – creates triangular & hexagonal faces = 3600°

It has: 4 triangular faces; 4 hexagonal faces; 8 total faces; 18 edges; 12 vertices

The net of the truncated tetrahedron:

A shallow truncation of the tetrahedron:

A full truncation (rectification) of the tetrahedron.  Notice it creates the octahedron.

## 4. Truncated octahedron – creates square & hexagonal faces = 7920°

It has: 6 square faces; 8 hexagonal faces; 14 total faces; 36 edges; 24 vertices

The vertices of the octahedron (shown in red) are truncated to form the truncated octahedron (shown below in green).

This is the only Archimedean solid that can fill space with identical copies of itself leaving no gaps. This means it tessellates in 3D space.

The net of the truncated octahedron:

## 5. Truncated cube – triangular & octagonal faces = 7920°

It has: 8 triangular faces; 6 octagonal faces; 14 total faces; 36 edges; 24 vertices

The Archimedean solid truncated cube is seen in the sequence below as the center shape:

The net of the truncated cube:

## 6. Truncated icosahedron – pentagonal & hexagonal faces = 20880°

It has: 12 pentagonal faces; 20 hexagonal faces; 32 total faces; 90 edges; 60 vertices

The net of the truncated icosahedron:

## 7. Truncated dodecahedron – triangular & decagonal faces = 20880°

It has: 20 triangular faces; 12 decagonal faces; 32 total faces; 90 edges; 60 vertices

The dodecahedron (below) is truncated to create the truncated dodecahedron (seen rotating in green).

Net of the truncated dodecahedron:

It is to be noted, “truncation alone is not capable of producing these solids, but must be combined with distorting to turn the resulting rectangles into squares.”3

There are other variations on truncation.  We discussed uniform truncation above.

Uniform truncation implies truncating until the original faces become regular polygons with equal side lengths.  In the below sequence uniform truncation is equal to ½ truncation.

There are also:

• ¼ truncation – cutting ¼ of the way
• ¾ truncation – cutting ¾ of the way
• Rectification – this reduces the original faces to points.

• Cantellation – cuts off edges instead of vertices. This involves removing the original edges and replacing them with rectangles.  Below the cube is cantellated to create the dodecahedron.

Credit: Frank Chester

Keep in mind, the importance of this is to show the sequence or progression of movements of Platonic solids.  They are never sitting still.  They are always oscillating and rotating, changing form and moving.

Look at the sequence below.  It shows the truncation sequence of a cube.  When the cube is rectified it becomes a cuboctahedron.  When further truncations are performed it moves through the entire sequence to become an octahedron – the dual of the cube.

Everything is in flux, and each solid can change into any other solid through a series of specific movements.

# Rectification

Each Platonic solid and its dual have the same rectified polyhedron.  Remember, rectification involves truncating the original face down to a point.

When rectified:

## A) A tetrahedron becomes the octahedron

You can clearly see how cutting the corners off of the tetrahedron down to its halfway point creates the octahedron.

## B) An octahedron and cube become a cuboctahedron

The cuboctahedron:

## C) An icosahedron and dodecahedron become an icosidodecahedron

The icosidodecahedron:

# Four Expansions (Explosions outward from the Center):

We have seen thus far, 7 of the 13 Archimedean solids.

Two are compound polyhedra and 5 are truncated polyhedra.

We will look at 4 more here, making 11.  The last two will be covered below.

Expansion, or cantellation, involves moving each face away from the center (by the same distance so as to preserve the symmetry of the Platonic solid) and taking the convex hull.

These have face planes in common with either the cube, octahedron and rhombic dodecahedron, or the icosahedron, dodecahedron and rhombic triacontahedron.

Therefore, ‘rhombi’ is in each name.

## 8. Rhombicuboctahedron = 7920° – triangle & square faces

It has: 8 triangular faces; 18 square faces; 26 total faces; 48 edges; 24 vertices

The net of the rhombicuboctahedron:

This is formed by exploding the faces of the cube or octahedron outwards until they are separated by an edge length, as seen below.

It can be used to make a structure similar to the jitterbug (the jitterbug of the cuboctahedron).

Applying a twist to this new structure creates the snub cube, as seen below.

## 9. Rhombicosidodecahedron = 20880° – triangle, square & pentagon faces

It has: 20 triangular faces; 30 square faces; 12 pentagonal faces; 62 total faces; 120 edges; 60 vertices

The net of the rhombicosidodecahedron:

This is formed by exploding the faces of the dodecahedron or icosahedron outwards until they are separated by an edge length, as shown below.

Applying a twist to this creates the snub dodecahedron, as seen below.

## 10. Great rhombicuboctahedron (a.k.a truncated cuboctahedron) = 16560°

It has: 12 square faces; 8 hexagonal faces; 6 octagonal faces; 26 total faces; 72 edges; 48 vertices

The net of the great rhombicuboctahedron:

This is formed by exploding the octagonal faces of the truncated cube or hexagonal faces of the truncated octahedron outwards until they are separated by an edge length.

## 11. Great rhombicosidodecahedron (a.k.a. truncated icosidodecahedron) = 42480°

It has: 30 square faces; 20 hexagonal faces; 12 decagonal faces; 62 total faces; 180 edges; 120 vertices

The net of the great rhombicosidodecahedron:

This is formed by exploding the decagonal faces of the truncated dodecahedron or the hexagonal faces of the truncated icosahedron outwards until they are separated by an edge length.

# Two Chiral Forms

The last two Archimedean solids both occur in left & right handedness.

Neither have mirror planes.

We briefly saw these above.  They are the:

## 12. Snub cube – octahedral symmetry = 7920°

It has: 32 triangular faces; 6 square faces; 38 total faces; 60 edges; 24 vertices

The net of the snub cube:

We saw above how it is formed by exploding the faces of the cube or octahedron outwards until they are separated by an edge length and then twisting.

## 13. Snub dodecahedron – icosahedral symmetry = 20880°

It has: 80 triangular faces; 12 pentagonal faces; 92 total faces; 150 edges; 60 vertices

The net of the snub dodecahedron:

We saw above how it is formed by exploding the faces of the dodecahedron or icosahedron outwards until they are separated by an edge length and then twisting.

Of all the Platonic & Archimedean Solids the snub dodecahedron is closest to the sphere.

These two solids “can be obtained by moving the faces of a cube and dodecahedron outward while giving each face a twist.  The resulting spaces are then filled with ribbons of equilateral triangles.”4

# Stellations

Stellations of solids involve extending the sides of some polygons until they meet again.  This forms points, creating solids that look like multi-faceted stars.

There are two possibilities when stellating:

• extending edges
• extending face planes

Amy Edmondson states, “Whereas truncation cuts off corners, ‘stellation’ is accomplished by the addition of a corner – imposing a shallow pyramid on a formerly flat face.  How many sides the pyramid will have is determined by the number of edges of the face to be stellated…Notice that—as required by their different faces—the cube’s superimposed pyramids have four sides, while the octahedron’s have three.”5

# Stellations of Regular Polygons

Two-dimensional polygons can also be stellated.  These result in the well-known star patterns such as pentagrams and hexagrams.

Pentagon –     1 stellation –    The pentagram

Hexagon –      1 stellation –    The hexagram or ‘Star of David’

Heptagon –     2 stellations – Heptagrams

Octagon –       2 stellations – Octagrams

Enneagon –    3 stellations – Enneagrams

Decagon –      3 stellations – Decagrams

# Stellations of Platonic Solids

Three of the regular polyhedra, or Platonic solids, can also be stellated.

The edges of the cube and tetrahedron once extended never meet, thus they have no stellations.

Tetrahedron – 0 stellations

Cube –  0 stellations

Octahedron – 1 stellation

• The stellated octahedron, Star Tetrahedron, or Stella Octangula

Dodecahedron – 3 stellations

• The Small stellated dodecahedron – 12 pyramids on the faces of the dodecahedron

The net of the small stellated dodecahedron:

• The Great dodecahedron – 30 wedges on the small stellated dodecahedron

The net of the great dodecahedron:

• The Great Stellated Dodecahedron – 20 spikes on the great dodecahedron

The net of the great stellated dodecahedron:

All three of these stellations are Kepler-Poinsot Solids.

Icosahedron – 58 stellations!

• These include 1 Kepler-Poinsot solid, 4 polyhedron compounds, and 1 dual polyhedron of an Archimedean solid.
• 32 of the stellations have full icosahedral symmetry.
• 27 are enantiomeric forms.  This means they are mirror images of other stellations.

The chart below shows all stellations including the original icosahedron (#1).

Rhombic Dodecahedron – 5 stellations

The rhombic dodecahedron is a Catalan solid, dual of the cuboctahedron (Archimedean solid) and an important shape in regards to galactic clustering.

Like the truncated octahedron, the rhombic dodecahedron tessellates – it fills 3D space with no gaps.

The first stellation of the rhombic dodecahedron is well known.  The last four are less known.

The first stellation is shown below:

It, like the rhombic dodecahedron, tessellates in 3D space:

The animation below shows the construction of a stellated rhombic dodecahedron by inverting the center-face pyramids of a rhombic dodecahedron.

# Kepler-Poinsot Polyhedra

“A Kepler-Poinsot polyhedron is any of four regular star polyhedra.  They may be obtained by stellating the regular convex dodecahedron and icosahedron.

The Kepler-Poinsot polyhedra exist in dual pairs:

• Small stellated dodecahedron and great dodecahedron

• Great stellated dodecahedron and great icosahedron.”6

These shapes were made famous by Johannes Kepler (1571-1630) and Louis Poinsot (1777-1859), however “most, if not all, of the Kepler-Poinsot polyhedra were known of in some form or other before Kepler.  A small stellated dodecahedron appears in a marble tarsia (inlay panel) on the floor of St. Mark’s Basiclica, Venice, Italy.  It dates from the 15th century and is sometimes attributed to Paolo Uccello.  In his Perspectiva corporum regularium (Perspectives of the regular solids), a book of woodcuts published in the 16th century, Wenzel Jamnitzer depicts the great dodecahedron and the great stellated dodecahedron.”7

These four solids are created by extending the edges of the dodecahedron and icosahedron.

This results in the first 2 stellations of the dodecahedron:

• small stellated dodecahedron
• great stellated dodecahedron

Each of these are made of 12 pentagram faces.

The stellated dodecahedron has 3 faces to every vertex.

The great stellated dodecahedron has 5 faces to every vertex.

Each has icosahedral symmetry.

These in turn have duals that are Kepler-Poinsot Polyhedra.

The dual of the small stellated dodecahedron is the great dodecahedron.

The dual of the great stellated dodecahedron is the great icosahedron.

The great dodecahedron and great icosahedron both have 5 faces to a vertex.

The Great dodecahedron has 12 pentagonal faces.  It is the 3rd stellation of the dodecahedron.

The Great Icosahedron has 20 triangular faces.  It is 1 of 58 stellations of the icosahedron.

# Faceting

“In geometry, faceting is the process of removing parts of a polygon, polyhedron or polytope without creating any new vertices.

New edges of a faceted polyhedron may be created along face diagonals or internal space diagonals.  A faceted polyhedron will have two faces on each edge and creates new polyhedra or compounds of polyhedra.

Faceting is the reciprocal or dual process to stellation.”8

The process of faceting produces the:

## 4 Kepler-Poinsot polyhedra (discussed above)

There are other compounds formed by the interrelationships among the nested platonic solids:      See page 157 in the Quadrivium.

Octahedra can be placed around a fixed icosahedron in 5 different ways.

This results in the compound of 5 octahedra (1 of 58 stellations of the icosahedron).

Cubes can be placed within a fixed dodecahedron in 5 different ways.

This results in the compound of 5 cubes.

A tetrahedron can be placed in a cube in 2 different ways (page 146).

This results in a compound of 2 tetrahedra – the star tetrahedron.

This star tetrahedron (stella octangula) is also a faceting of the cube.

Replace each cube in the 5 cube dodecahedron with 2 tetrahedra.

This results in the compound of 10 tetrahedra (another stellation of the icosahedron).

Replace 5 of the tetrahedra from the 10 tetrahedra compound.

This results in the compound of 5 tetrahedra (another stellation of the icosahedron).

A dodecahedron can be placed around a fixed cube.

This results in the compound of 2 dodecahedra.

The octahedron and icosahedron pair can be added together.

This results in the compound of 2 icosahedra.

# Chiral or ‘handedness’

The compound of 5 tetrahedra above can be:

• right-handed (dextro)
• left-handed (laevo)

Superimposed on one another they are each other’s ‘enantiomorphs’.  This means they are mirror images.

Recall that 27 of the stellations of the icosahedron are enantiomorphs.

Chiral – the property of a polyhedra having left and right handedness.

Recall that 2 of the Archimedean solids are chiral – the snub cube and snub dodecahedron.

# The Catalan Solids: Archimedean Duals

The Catalan solids were first described as a group by Eugene Catalan (1814-1894).

Perpendicular lines are extended from their edge midpoints, tangential to the Solid’s midsphere.

The lines are the dual’s edges.

The points where they first intersect each other are its vertices.

The duals have one type of face but different types of vertices.

## 1. Triakis Tetrahedron – 2160°

It has: 12 faces; 18 edges; 8 vertices

• triakis tetrahedron: truncated tetrahedron

Net of the Triakis Tetrahedron:

## 2. Tetrakis Hexahedron – 4320°

It has: 24 faces; 36 edges; 14 vertices

• tetrakis hexahedron : truncated octahedron

Net of the Tetrakis Hexahedron:

## 3. Rhombic Dodecahedron – 2160°

It has: 12 faces; 24 edges; 14 vertices

• rhombic dodecahedron : cuboctahedron

• Exploding either of these duals creates an equal-edged convex polyhedron with 50 faces.
• The rhombic dodecahedron can tessellate 3D space.
• The only other solids to do this are the cube and truncated octahedron.

Net of the Rhombic Dodecahedron:

## 4. Triakis Octahedron – 4320°

It has: 24 faces; 36 edges; 14 vertices

• Triakis octahedron : truncated cube

Net of the Triakis Octahedron:

## 5. Trapezoidal Icositetrahedron (a.k.a. Deltoidal Icositetrahedron) – 8640°

It has: 24 faces; 48 edges; 26 vertices

• trapezoidal icositetrahedron : rhombicuboctahedron

Net of the Trapezoidal Icositetrahedron:

## 6. Disdyakis Dodecahedron – 8640°

It has: 48 faces; 72 edges; 26 vertices

• disdyakis dodecahedron : great rhombicuboctahedron

Net of the Disdyakis Dodecahedron:

## 7. Pentagonal Icositetrahedron – 12960°

It has: 24 faces; 60 edges; 38 vertices

• pentagonal icositetrahedron : snub cube

Net of the Pentagonal Icositetrahedron:

## 8. Pentakis Dodecahedron – 10800°

It has: 60 faces; 90 edges; 32 vertices

• pentakis dodecahedron : truncated icosahedron

Net of the Pentakis Dodecahedron:

## 9. Rhombic Triacontahedron – 10800°

It has: 30 faces; 60 edges; 32 vertices

• rhombic triacontahedron : icosidodecahedron

• Exploding either of these duals creates a 122 face polyhedra.
• The rhombic tricontahedron is made of 30 φ:1 diamonds.

Net of the Rhombic Triacontahedron:

## 10. Triakis Icosahedron – 10800°

It has: 60 faces; 90 edges; 32 vertices

• Triakis icosahedron: truncated dodecahedron

Net of the Triakis Icosahedron:

## 11. Trapezoidal Hexecontahedron (a.k.a. deltoidal hexecontahedron) – 21600°

It has: 60 faces; 120 edges; 62 vertices

• trapezoidal hexecontahedron : rhombicosidodecahedron

Net of the Trapezoidal Hexecontahedron:

## 12. Disdyakis Triacontahedron – 21600°

It has: 120 faces; 180 edges; 62 vertices

• disdyakis triacontahedron : great rhombicosidodecahedron

Net of the Disdyakis Triacontahedron:

## 13. Pentagonal Hexecontahedron – 32400°

It has: 60 faces; 150 edges; 92 vertices

• pentagonal hexecontahedron : snub dodecahedron

Net of the Pentagonal Hexecontahedron:

 Catalan Solid Archimedean Dual Faces Edges Vertices Sum of Angles Triakis Tetrahedron Truncated Tetrahedron 12 18 8 2160 Tetrakis Hexahedron Truncated Octahedron 24 36 14 4320 Triakis Octahedron Truncated Cube 24 36 14 4320 Pentakis Dodecahedron Truncated Icosahedron 60 90 32 10800 Triakis Icosahedron Truncated Dodecahedron 60 90 32 10800 Rhombic Dodecahedron Cuboctahedron 12 24 14 2160 Trapezoidal Icositetrahedron Rhombicuboctahedron 24 48 26 8640 Disdyakis Dodecahedron Great Rhombicuboctahedron 48 72 26 8640 Rhombic Triacontahedron Icosidodecahedron 30 60 32 10800 Trapezoidal Hexecontahedron Rhomb-icosidodecahedron 60 120 62 21600 Disdyakis Triacontahedron Great Rhomb-icosidodecahedron 120 180 62 21600 Pentagonal Icositetrahedron Snub Cube 24 60 38 12960 Pentagonal Hexecontahedron Snub Dodecahedron 60 150 92 32400 Sum of All 151,200

The Platonic Dual Pairs (star tetrahedron, cuboctahedron, icosidodecahedron) define the face diagonals of these rhombic polyhedra.

They are in ratios of:

• √2 for the rhombic dodecahedron (√2:1)

• phi for the rhombic triacontahedron

Interestingly, Kepler noticed bees terminate their hexagonal honeycomb cells with 3 such √2 rhombs (from the rhombic dodecahedron).

Kepler wrote, “If you should ask the geometers on what plan the cells of bees are built, they will reply, on a hexagonal plan. The answer is clear from a simple look at the openings or entrances, and the sides that form the cells. Each cell is surrounded by six others, and is divided from the next by a shared side. But if you observe the bottom of each cell, you will notice that it slopes into an obtuse angle formed by three planes. This bottom (which you might call the “keel”) is joined to the six sides of the cell by six other angles, three higher ones that are trilateral, just like the bottom angle of the keel, and three lower ones, in between, that are quadrilateral. It can also be observed that the cells are arranged in two layers, with the openings facing opposite directions; the backs adjoining one another and closely packed; and the corners of each keel in one layer fitted into the three corners of the three keels in the other…. The three planes of the keel are identical to one another, and their shape is what geometers call a rhombus.”9

# Polytopes

“In elementary geometry, a polytope is a geometric object with ‘flat’ sides.  It is a generalization in any number of dimensions of the three-dimensional polyhedron.”10

This is a mathematically complex subject.  We will only touch upon it here.

There are 6 regular 4-dimensional generalizations of polyhedra.  4-polytopes are four-dimensional analogs of a Platonic solid.

These were proved by Ludwig Schlafi (1814-1895) and include:

## 5-cell of tetrahedra

• The 5-cell is analogous to the tetrahedron in three dimensions and the triangle in two.
• It is self-dual, and its vertex figure is a tetrahedron.
• The 5-cell is essentially a 4-dimensional pyramid with a tetrahedral base.11

## 8-cell of cubes (Tesseract)

• The rhombic dodecahedron is a 3D shadow of the 4D tesseract.
• It is analogous to the hexagon as a 2D shadow of the 3D cube.

• In a cube 2 squares meet at every edge.
• In a tesseract 3 squares meet at every edge.

• Each vertex of a tesseract is adjacent to four edges.
• The vertex figure is a regular tetrahedron.  Its dual is the 16-cell.
• In all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices.

## 16-cell of tetrahedra

• The 16-cell is bounded by 16 cells, all of which are regular tetrahedra.
• It has 32 triangular faces, 24 edges, and 8 vertices.
• The 24 edges bound 6 squares lying in the 6 coordinate planes.
• Its vertex figure is a regular octahedron.
• There are 8 tetrahedra, 12 triangles, and 6 edges meeting at every vertex.
• Its edge figure is a square.  There are 4 tetrahedra and 4 triangles meeting at every edge.
• The 16-cell can be dissected into two octahedral pyramids.12

## 24-cell of octahedra

• The 24-cell is composed of 24 octahedral cells with 6 meeting at each vertex, and 3 at each edge.
• Together they have 96 triangular faces, 96 edges, and 24 vertices.  It is self-dual.13

## 120-cell of dodecahedra

• The 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex.
• It is the 4D analog of the dodecahedron.
• There are 120 cells, 720 pentagonal faces, 1200 edges and 600 vertices.
• There are 4 dodecahedra, 6 pentagons, and 4 edges meeting at every vertex.
• There are 3 dodecahedra and 3 pentagons meeting every edge.
• The vertex figure is a tetrahedron.  Its dual is the 600-cell.14

## 600-cell of tetrahedra

• The 600-cell is the 4D analog of the icosahedron.
• It has five tetrahedra meeting at every edge.
• It is composed of 600 tetrahedral cells with 20 meeting at each vertex.
• Together they form 1200 triangular faces, 720 edges, and 120 vertices.
• The edges form flat regular decagons.  Each vertex is a vertex of 6 such decagons.15

# 5D Polytopes

Five-dimensional polytopes were also proved by Schlafi.

There are three main classes of regular polytope which occur in any number n of dimensions:

• simplex – including the equilateral triangle and regular tetrahedron
• hypercube – including the square and cube
• orthoplex – including the square and regular octahedron

1. Wolfram Math World, Dual Polyhedron, http://mathworld.wolfram.com/DualPolyhedron.html
2. Edmondson, Amy, A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller, Burkhauser Boston, 1987
3. http://mathworld.wolfram.com/ArchimedeanSolid.html
4. ibid.
5. Edmondson, Amy, A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller, Burkhauser Boston, 1987
6. https://en.wikipedia.org/wiki/Kepler%E2%80%93Poinsot_polyhedron
7. ibid.
8. https://en.wikipedia.org/wiki/Faceting
9. Matematicas Visuales, http://www.matematicasvisuales.com/english/html/geometry/rhombicdodecahedron/honeycomb.html
10. https://en.wikipedia.org/wiki/Polytope
11. https://en.wikipedia.org/wiki/5-cell
12. https://en.wikipedia.org/wiki/16-cell
13. https://en.wikipedia.org/wiki/24-cell
14. https://en.wikipedia.org/wiki/120-cell
15. https://en.wikipedia.org/wiki/600-cell