Ambo Polyhedron

Kepler and the rectified polyhedra = ambo of Conway polyhedra

1. The geometric operation of "rectification" of a polyhedron

In modern geometry (especially since the work of Coxeter and Conway 's notation ), "rectifying" a polyhedron (P) is an operation that produces a new polyhedron ( operatorname { rect }(P) ), whose vertices are the midpoints of the edges of (P).

How is it built?

1. “Take the midpoints of each edge” of the original polyhedron.

2. “These points become the vertices” of the new polyhedron.

3. “The original faces” are reduced to new faces, with the same number of sides but smaller (they are “truncated to half the edges”).

4. The “original vertices” disappear; instead, for each vertex of the original polyhedron, a new face appears. This new face has as many sides as edges converged at the original vertex (i.e., a face of type (n) -gonal where (n) is the degree of the vertex).

Therefore, rectification transforms:

- Faces with (n) sides → Faces with (n) sides (smaller).

- Vertices of degree (m) → Faces of (m) sides.

This preserves the original symmetry.

Key properties

Rectifying a polyhedron and its dual yields the same result. For example, rectifying the cube and octahedron (which are duals of each other) produces the cuboctahedron . Rectifying the dodecahedron and icosahedron produces the icosidodecahedron .

- It is equivalent to a “complete truncation” (to the midpoint of the edges).

- In Conway terms , the operation is called “ambo” (a)

Examples

| Original polyhedron | Rectified polyhedron | Rectified faces |

| Tetrahedron | Octahedron (regular) | 8 triangles ( 3 for each vertex of the tetrahedron + 4 triangles from the original faces? ) *Note: the rectified tetrahedron is the octahedron; its faces are the 4 triangles from the original faces and the 4 triangles from the vertices, totaling 8. |

| Cube | Cuboctahedron | 8 triangles (from the 8 vertices of the cube) + 6 squares (original faces reduced) |

| Octahedron | Cuboctahedron | 6 squares (from the 6 vertices of the octahedron) + 8 triangles (original faces reduced) → The same cuboctahedron |

| Dodecahedron | Icosidodecahedron | 20 triangles (from the 20 vertices) + 12 pentagons (original faces) |

| Icosahedron | Icosidodecahedron | 12 pentagons (from the 12 vertices) + 20 triangles (original faces) → Same solid |

2. Kepler and the rectified polyhedra

Although the word "rectification" in the modern sense does not appear in his texts, the geometric process I have described is exactly what Johannes Kepler explored in his book " Harmonices Mundi " (1619, *The Harmonies of the World*), Books I and II.

Rediscovery of Archimedean solids

The 13 Archimedean solids (convex polyhedra with regular faces of two or more types, identical vertices) had been described by Archimedes, but his work was lost and references were only known in Pappus . “Kepler reconstructed them all” and presented them systematically for the first time in the modern era, with names that in many cases have endured.

For him they were “perfect figures of the second order” (the Platonic ones were of the first order). Kepler showed how to obtain them from the Platonic ones through truncations, cuts at the vertices and edges.

Among those 13, two are precisely the ones we now call "rectified":

- The “ cuboctahedron ”: Kepler called it *a truncated cube and a truncated octahedron at the same time*, because he observed that truncating the cube at the midpoints of its edges and truncating the octahedron in the same way yields the same solid. He considered it a fusion of both truncations.

- The “ icosidodecahedron ”: Same relationship between dodecahedron and icosahedron. Kepler explicitly describes it as a body that can be generated “by truncating the dodecahedron or the icosahedron until the edges become points”, that is, until the midpoints.

Keplerian construction step by step (example cube → cuboctahedron )

Kepler in * Harmonices Mundi *, Book II, proposition XXVIII, describes the truncation of the cube:

1. On each edge of the cube mark two points that divide it according to the extreme and average (golden) proportion or, in the limiting case, exactly in the middle.

2. Cut planes that pass through those points, eliminating the corners of the cube.

3. If you cut the edge in half, the original square faces become smaller squares, and an equilateral triangle appears at each former vertex. The result is a solid with 6 squares and 8 triangles: the cuboctahedron .

Kepler did not limit himself to construction; he also “calculated metric properties” (edges, radii of inscribed and circumscribed spheres) as a function of the original radius, and explored how these solids could be inscribed in one another.

Keplerian nomenclature and notation

Kepler used descriptive Latin names:

- “ Cuboctahedron ” (from * cubus * + * octahedron *) → * cuboctahedron *.

- “ Icosidodecahedron ” (* icosadodecahedron * + * dodecahedron *) → * icosidodecahedron *.

These names were adopted by later tradition and are the ones we use today.

Star polyhedra and the rectification operation

Kepler also discovered the two "regular stellated polyhedra" (the * stella octangula * and what are now called the Kepler- Poinsot solids: the small stellated dodecahedron and the large stellated dodecahedron). Although they are not directly rectified, their formation involved faceting and face extension processes , which in modern theory are related to the operations of rectification and truncation in symmetry spaces.

When in modern geometry we talk about "rectifying a polyhedron", the result is precisely the Archimedean solids that Kepler rediscovered and described with the method of complete truncation to the midpoint

In expanded summary

- “Rectification (modern)”: an operation that creates a polyhedron whose vertices are the midpoints of the edges of the original. It transforms original faces into reduced versions and vertices into new faces. It is a particular case of total truncation.

Kepler reconstructed the 13 Archimedean solids , including the cuboctahedron and the icosidodecahedron , which are exactly the rectified versions of the cube/octahedron and the dodecahedron/icosahedron. He obtained them through symmetrical truncations that coincide with modern rectification.

- His treatise * Harmonices Mundi * provides the first classification, nomenclature and metric study of these polyhedra, laying the foundations for the later development of the geometry of operators on polyhedra.

The table with the “13 Archimedean solids ” in Conway notation and Kepler's description of each one. I will take as a starting point the Platonic solids: T (tetrahedron), C (cube), O (octahedron), D (dodecahedron), I (icosahedron).

The 13 Archimedean solids according to Kepler and in Conway notation

Archimedean solid (modern name) | Name given by Kepler | Conway notation | How Kepler constructed/described it |

| 3 | “Truncated octahedron” | * Octaëdron truncum * | “ tO ” | Truncate the vertices of the octahedron; the triangular faces become hexagons and the truncated vertices become squares. Kepler noted that it is the only Archimedean tessellation that tiles space. |

| 5 | “Truncated icosahedron” | * Icosaëdron truncum * | “ tI ” | Truncating the icosahedron; the triangular faces become regular hexagons and the cut vertices become pentagons. Kepler recognized it, although it is more famous for its association with the fullerene . |

| 6 | “ Cuboctahedron ” | * Cuboctahedron * | “ aC ” (= aO ) | Truncate the cube *exactly at the midpoint of its edges* (or do the same with the octahedron). The square faces are reduced and the vertices become triangles. Kepler observed that this is the point of intersection between the cube and the octahedron.

| 8 | “ Rhombicuboctahedron ” (small) | * Rhombicuboctahedron * | “ eC ” (= eO ) | “Expand” the cube: separate each face, add a rectangle for each edge and a square for each original vertex. Kepler described it as a smooth truncation, where the faces are not completely removed but belts of squares are added. |

| 9 | “ Rhombicosidodecahedron ” (small) | * Rhombicosidodecahedron * | “ eD ” (= eI ) | Expansion of the dodecahedron: each pentagon moves further away, each edge becomes a square, and each vertex becomes a triangle. Kepler considered it an “even more perfect” figure because of its rich symmetry. |

| 10 | “Great rhombicuboctahedron ” ( truncated cuboctahedron ) | * Truncum Cuboctaëdron * | “ bC ” (= bO ) | Truncate the cuboctahedron (or apply a bevel to the cube). Octagons, hexagons, and squares result. Kepler says it is a deeper truncation of the cube, beyond the cuboctahedron . |

| 11 | “Great rhombicosidodecahedron ” ( truncated icosidodecahedron ) | * Truncum Icosidodecahedron * | “ bD ” (= bI ) | Similarly, truncating the icosidodecahedron yields decagons, hexagons, and squares. Kepler obtained it by “extreme corner cutting”. |

| 12 | “Snub cube” | * Cubus simus * | “ sC ” | It is obtained by “twisting” the faces of the cube and filling the gaps with triangles. Kepler described it as a solid where the original faces are rotated and each former edge becomes two triangles. It is chiral ; it exists in two mirror-image forms.

| 13 | “Snub dodecahedron” | * Dodecahedron simum * | “ sD ” | Same as the snub cube but derived from the dodecahedron. Kepler considered it legitimate as a second-order figure, although its construction is more complex. |

Important observations

- “ Conway notation ”: the letters indicate operators applied to an initial Platonic solid:

- “t” = truncate (cut vertices)

- “a” =ambo / rectify (midpoints of the edges)

- “b” = bevel ( to bevel , is equivalent to t ∘ a )

- “e” = expand (separate faces and add belts of squares or triangles)

- “s” = snub (twist and fill with triangles)

- The equality between seeds (e.g., aC = aO ) reflects duality: cube and octahedron are dual, and when rectified they give the same solid; the same applies to dodecahedron and icosahedron.

The names “ rhombi- ”, “ cuboctahedron ”, “ icosidodecahedron ”, and “ simus ” are exactly those coined by Kepler in Harmonices Mundi (1619) and have survived to this day. His classification is the first complete one and marked the rebirth of the geometry of semiregular polyhedra .

Kepler did not use the word "rectification" for polyhedra, but his description of "truncating to half the edges" is "exactly" Conway 's "ambo(a)" operation , the current rectification. Therefore, the Archimedean solids that we now call *rectified* ( cuboctahedron and icosidodecahedron ) are the Keplerian "ambos" .

Basic meaning: In its simplest form, the Ambo process is an operation that takes a polyhedron and produces a new polyhedron whose vertices are the midpoints of the edges of the original polyhedron. Imagine each edge of a polyhedron. Now mark its midpoint. If we connect all these midpoints in a specific way (explained later), we will obtain a new polyhedron: the Ambo of the original.

Step-by-step process:

To apply the Ambo process to a polyhedron, the following steps are followed:

Identify the new vertices: The new vertices are exactly the midpoints of all the edges of the original polyhedron.

Connecting the vertices: To form the faces of the new polyhedron, the new vertices are connected according to this rule: Two new vertices (midpoints) are connected to form an edge Ambo if the original edges from which they come shared the same face in the original polyhedron.

Resulting polyhedron type: The ambo polyhedron is always a uniformly grouped vertex polyhedron, meaning that all its vertices are identical by grouping.

Relationship to the original: The ambo has one face for each face of the original polyhedron. However, if the original face had n sides, the corresponding new face will also have n sides, but will be smaller and "cut off" (remaining in the center of the original face).

The ambo has one face for each vertex of the original polyhedron. Each vertex of the original, where there were m edges, becomes a new m-sided face in the ambo.

In notation: Ambo(Ambo(P)) is similar to P.

Practical examples:

Ambo of the cube. Original polyhedron: Cube (6 square faces, 8 vertices, 12 edges). Process: The midpoints of the 12 edges of the cube are taken.

The resulting polyhedron is a cuboctahedron . It has 12 vertices (one for each edge of the cube). It has 14 faces: 6 squares (from the 6 faces of the cube) and 8 triangles (from the 8 vertices of the cube). All its vertices are identical: at each one, a square and two triangles meet.

2. Ambo of the Octahedron Original Polyhedron: Octahedron (8 triangular faces, 6 vertices, 12 edges). Process: The midpoints of the 12 edges of the octahedron are taken. Resulting ambo polyhedron: Again, a cuboctahedron .

This shows that the cube and the octahedron (which are dual polyhedra) produce the same polyhedron.

3. Ambo of the Dodecahedron

Original polyhedron: Dodecahedron (12 pentagonal faces). Resulting polyhedron: An icosidodecahedron . It has 30 vertices (corresponding to the 30 edges of the dodecahedron). It has 32 faces: 12 pentagonal and 20 triangular.

Relationship with other operators and applications:

The Ambo process is fundamental in polyhedron theory and is one of the Conway operators , named after the mathematician John Horton Conway .

These operators (such as "rectify," "truncate," "expand," etc.) allow the generation of a vast family of polyhedra from the Platonic solids. The Ambo process is, in fact, identical to the rectification process, where the vertices of the original polyhedron are "cut" until the cutting plane exactly reaches the midpoints of all the edges. Its best-known application is the first step in constructing a soccer ball (a truncated icosahedron) or fullerenes (carbon molecules), starting with an icosahedron and applying operations that include the Ambo/rectification process.

Summary

In essence, the ambo is a geometric transformation process that: Reduces the symmetry of the polyhedron to vertices that are uniform with each other.

It creates a polyhedron intermediate between a polyhedron and its dual. It increases the number of faces (by adding the faces and vertices of the original). It is a fundamental tool for generating and classifying complex polyhedra from simpler ones.

The origin of the word "ambo" in this geometric context is very specific and is directly linked to the work of the mathematician John Horton Conway .

Direct origin: John Horton Conway. The term "ambo" was coined by the prolific and brilliant mathematician John Horton Conway (1937–2020) as part of his system of polyhedral operators. Conway , known for his playful and creative style of naming mathematical concepts, developed a notation where simple letters or syllables represent fundamental operations that can be applied to a polyhedron to generate a new one.

Etymology and meaning of the term

The word "ambo" comes from Latin, where it means "both." Why did Conway choose this term?

The reason is deeply descriptive and captures the very essence of the operation: the polyhedron resulting from the "ambo" operation can be considered to belong to both polyhedra: the original and its dual.

Let's take the classic example: the dual of a cube is a cuboctahedron , and the dual of an octahedron (which is the cube's dual) is also a cuboctahedron . The resulting cuboctahedron is, therefore, a kind of "hybrid" or "intermediate state" that shares characteristics of both parent polyhedra (the cube and the octahedron). It is an almost regular polyhedron that sits right in the middle of the dual pair. In essence, the "dual" polyhedron is the "child" that equally inherits the properties of both parents (the solid and its dual).

In notation: Ambo(Ambo(P)) is similar to P.

Relationship with other operators and applications: The ambo process is fundamental in polyhedron theory and is one of the Conway operators , named after the mathematician John Horton Conway . These operators (such as "rectify," "truncate," "expand," etc.) allow the generation of a vast family of polyhedra from the Platonic solids. The ambo is, in fact, identical to the rectification process, where the vertices of the original polyhedron are "cut" until the cutting plane exactly reaches the midpoints of all the edges.

Most well-known application: It is the first step in the construction of a soccer ball (truncated icosahedron) or fullerenes (carbon molecules), starting from an icosahedron and applying operations that include ambo/rectification.

Summary: In essence, the ambo is a geometric transformation process that:

It reduces the symmetry of the polyhedron to vertices that are uniform with respect to each other. It creates a polyhedron intermediate between a polyhedron and its dual. It increases the number of faces (by adding the faces and vertices of the original). It is a fundamental tool for generating and classifying complex polyhedra from simpler ones.

The origin of the word "ambo" in this geometric context is very specific and is directly linked to the work of the mathematician John Horton Conway .

Direct origin : John Horton Conway. The term "ambo" was coined by the prolific and brilliant mathematician John Horton Conway (1937-2020) as part of his system of polyhedral operators. Conway , known for his playful and creative style of naming mathematical concepts, developed a notation where simple letters or syllables represent fundamental operations that can be applied to a polyhedron to generate a new one.

Etymology and meaning of the term

The word "ambo" comes from Latin, where it means "both." Why did Conway choose this word? The reason is profoundly descriptive and captures the very essence of the operation: the polyhedron resulting from the ambo operation can be considered to belong to both polyhedra: the original and its dual. Let's take the classic example: the ambo of a cube is a cuboctahedron , and the ambo of an octahedron (which is the dual of the cube) is also a cuboctahedron . The resulting cuboctahedron is, therefore, a kind of "hybrid" or "intermediate state" that shares characteristics of both parent polyhedra (the cube and the octahedron).

It is a quasi-regular polyhedron that lies precisely in the middle of the dual pair. Essentially, the ambo polyhedron is the "child" that equally inherits the properties of "both" parents (the solid and its dual). In Conway notation , operators are applied as a string of letters to the description of a seed polyhedron (e.g., "T" for tetrahedron, "C" for cube, "O" for octahedron, "D" for dodecahedron, "I" for icosahedron). The operator "a" (ambo) is one of the most fundamental. For example: aC means "ambo of cube," which is the cuboctahedron . aO means "ambo of octahedron," which is also the cuboctahedron . aD (ambo of dodecahedron) produces the icosidodecahedron .

Summary : Creator : John Horton Conway. Original language: Latin. Literal meaning: "Both." Reason for its name: Because the resulting polyhedron acts as an intermediate or common point between a polyhedron and its dual, sharing properties of both. It is a perfect example of Conway 's genius : a short, memorable, and deeply descriptive term that encapsulates the essence of a complex geometric transformation.

Comparative Summary Table

	Regular	Uniform Polyhedra	Johnson polyhedra
Faces	One type of regular face	Regular faces (different)	Regular faces (different)
Vertices	Only one type of vertex	Vertex-transients	Non-transitive vertices
Symmetry	The greatest symmetry	High global symmetry. All vertices are identical.	Low or local symmetry. The vertices are of different types.
Number	5	75 (with infinite series of prisms/ antiprisms ).	92 (finite)
They include	Platonic Solids	Platonic solids, Archimedean solids , prisms and antiprisms	Pyramids, domes, roundabouts and combinations thereof.

Partial ambo of the faces of the cube

Partial ambo of the faces of an octahedron

Together they form the faces of a cuboctahedron (cube and octahedron are duals)

Ambo of the faces of a dodecahedron

Ambo of the faces of an icosahedron

If we put them together we get the icosidodecahedron (the icosahedron is the dual of the dodecahedron)