Acrohedra

What is an acrohedron?  An acrohedron is a polyhedron containing acrons (or vertices).  Acron stems from 'acros', Greek - summit, as in Acropolis.  The definition so far is not too useful.  However if we define an XYZ-acron (or an 'X-Y-Z') as being a vertex surrounded in sequence by an X-gon, Y-gon and Z-gon then we can define an XYZ-acrohedron as being a polyhedron containing at least one XYZ-acron.  A cube for example can be regarded as a 444-acrohedron.

The image above shows a 9-4-3 acrohedron

Trihedral Acrons

Starting with trihedral acrons, or acrons surrounded by three polygons, we can generate a list of potential acrons.  To make the list manageable, we introduce two limitations:

1. We shall exclude star polygons

2. We shall only consider acrons where there is a positive angular deficit, i.e. the sum of the internal angles of the constituent polygons is less than 360 degrees.

We shall temporarily introduce a third limitation; that is for the time being we shall only consider polygons that occur in the uniform polyhedra or in the Johnson Solids. This reduces the polygons under consideration to the triangle, square, pentagon, hexagon, octagon and decagon.  I have heard these polygons referred to (with tongue in cheek) as the 'polydronic' polygons after the Polydron polyhedral construction kits.  Other polygons then become termed 'apolydronic'.

The above limitations leave a list of 34 possibilities (below).

For each acron we can now search for an acrohedron, specifically we can search for the minimal acrohedron.  This is the polyhedron containing the requisite acron that has the minimum number of faces, edges or vertices.  Interestingly not all acrons are represented in the uniform polyhedra or in the Johnson Solids. Some like the 5-4-3 have brought to light lesser-known polyhedra like the Stewart G3 or have required a specific construction, such as Professor Conway's solution for the 6-5-4.  Others, like the 5-5-4, have so far eluded solution.

In certain cases the minimum acrohedron is not clear-cut.  Cases such as the 5-5-5 yield a different solution (the Stewart G3) for the minimum number of vertices then for the minimum number of faces (the dodecahedron).  In cases such as this both polyhedra are listed.

In other cases the minimum acrohedron contains coplanar faces (e.g. the 6-5-4), or is self-intersecting (e.g. the 10-5-3).  For cases such as this both the minimal (coplanar) solution and the minimal (non-coplanar) solution and/or the minimal self-intersecting and the minimal non-self-intersecting polyhedra are listed.

A table of dihedral angles is available here.   

A list of the trihedral acrons follows.  Note the convention that in moving from X to Y to Z, one moves from the largest polygon to the smallest.
 
trihedral
acron
Deficit (degrees) V, E, F Polyhedron
3-3-3 180 4, 6, 4 tetrahedron  see also n-n-3 acrons
4-3-3 150 5, 8, 5 square pyramid (J01)
4-4-3 120 6, 9, 5 triangular prism  see also n-n-3 acrons
4-4-4 90 8, 12, 6 cube
5-3-3 132 6, 10, 6 pentagonal pyramid (J02)
5-4-3 102 10, 18, 9

10, 20, 12

Small Ditrigonal Icosidodecahedron facet (self intersecting) - R.Klitzing  click here for more these 

Stewart m*

5-4-4 63 10, 15, 7 pentagonal prism
5-5-3 84 9, 15, 8 tridiminished icosahedron (J63)  see also n-n-3 acrons
5-5-4 54 none known view some near misses
5-5-5 36 13, 24, 13

20, 30, 12

Stewart G3   click here for more

dodecahedron

6-4-3 90 9, 15, 8 triangular cupola (J03)
6-4-4 60 12, 18, 8 hexagonal prism
6-5-3 72 none known view some near misses
6-5-4 42 27, 42, 17

18, 36, 20

22, 48, 28

Rhombidodecadodecahedron facet (self intersecting) - R.Klitzing  click here for more on all three 

Conway 6-5-4 (coplanar faces) 

Conway 6-5-4 (no coplanar faces)

6-5-5 24 none known view some near misses
6-6-3 60 12, 18, 8 truncated tetrahedron  see also n-n-3 acrons
6-6-4 30 24, 36, 14 truncated octahedron
6-6-5 12 60, 90, 32 truncated icosahedron
8-4-3 75 12, 20, 10 square cupola (J04)
8-4-4 45 16, 24, 10 octagonal prism
8-5-3 57 none known
8-5-4 27 none known
8-5-5 9 none known
8-6-3 45 none known
8-6-4 15 48, 72, 26 truncated cuboctahedron
8-8-3 30 24, 36, 14 truncated cube  see also n-n-3 acrons
10-4-3 66 15, 25, 12 pentagonal cupola (J05)
10-4-4 36 20, 30, 12 decagonal prism
10-5-3 48 20, 32, 14

20, 35, 17

icosidodecahedron facet (self-intersecting)

pentagonal rotunda (J06)

10-5-4 18 45, 75, 32 tridiminished rhombicosidodecahedron (J83)
10-6-3 36 40, 70, 28

28, 56, 30

30, 62, 34

Small icosicosidodecahedron facet (self intersecting) - R.Klitzing  click here for more on all three

Conway 10-6-3 (coplanar faces)   click here for info

Conway 10-6-3 (non-coplanar faces)

10-6-4 6 120, 180, 62 truncated icosidodecahedron
10-8-3 21 none known
10-10-3 12 60, 90, 32 truncated dodecahedron  see also n-n-3 acrons

Removing now our limitation to polydronic polygons, we can now consider the remaining apolydronic polygons.  Apart from the N-4-4, which for all N yields the N-gonal prism (e.g. 7-4-4), acrons containing apolydronic polygons such as the heptagon seem to only rarely yield acrohedra.  The list below shows all apolydronic trihedral acrons known to date to yield solutions.  Click here for a full list of potential acrons.
 
trihedral
acron
Deficit (degrees) V, E, F Polyhedron
7-4-3 81 3/7 20, 40 ,22 McNeill 7-4-3   click here for more
7-6-4 21 3/7 71, 140, 70

84, 182, 99

McNeill 7-6-4 (self-intersecting - coplanar faces)   click here for info

McNeill 7-6-4 (self-intersecting - non-coplanar faces)

7-7-3 42 6/7 28, 49, 22 Mason Green's Small Supersemicupola (self-intersecting)   click here for more, see also n-n-3 acrons
9-4-3 70 26, 52, 28 McNeill 9-4-3
11-4-3 62 8/11 32, 64, 34 McNeill 11-4-3

Flat acrons, where the deficit is zero, always yield solutions.  Simply join two such acrons together by a ring of squares.  Click here for a full list of flat acrons.

A special case of the tri-hedral acron is the n-n-3 acron.  Click here for more information on these.

Tetrahedral Acrons

Tetrahedral acrons have four surrounding polygons.  In order to keep the list manageable we add to our list of restrictions as follows: the acron itself must be convex, or non-reflexive and must not be crossed.  Acrons such as the crossed 4-3-4-3 occurring in the tetrahemihexahedron will not be considered here.

In considering the tetrahedral acrons we must differentiate between the X-Y-Z-Z and the X-Z-Y-Z, as the sequence of polygons is relevant.

A list of the polydronic tetrahedral acrons follows:
 
tetrahedral
acron
Deficit (degrees) V, E, F Polyhedron
3-3-3-3 120 5, 8, 5 square pyramid (J01)
4-3-3-3 90 8, 16, 10 square antiprism
4-3-4-3 60 8, 14, 8 gyrobifastigium (J26)
4-4-3-3 60 7, 12, 7 elongated triangular pyramid (J07)
4-4-4-3 30 12, 20, 10 square cupola (J04)
5-3-3-3 72 9, 15, 8 tridiminished icosahedron (J63)
5-3-4-3 42 14, 26, 14 bilunabirotunda (J91)
5-3-5-3 24 14, 26, 14 bilunabirotunda (J91)
5-4-3-3 42 11, 18, 9 augmented pentagonal prism (J52)
5-4-3-4 12 15, 25, 12 pentagonal cupola (J05)
5-4-4-3 12 30, 52, 24

30, 55, 27

prism augmented icosidodecahedron facet (self-intersecting)

elongated pentagonal rotunda (J21)

5-5-3-3 24 10, 18, 10 augmented tridiminished icosahedron (J64)
6-3-3-3 60 12, 24, 14 hexagonal antiprism
6-3-4-3 30 15, 27, 14 augmented truncated tetrahedron (J65)
6-3-5-3 12 38, 81, 45

40, 87, 49

Conway 6-3-5-3 (coplanar faces) click here for more

Conway 6-3-5-3 (non-coplanar faces)

6-4-3-3 30 13, 22, 11 augmented hexagonal prism (J54)
6-5-3-3 12 28, 46, 20

19, 40, 23

23, 52, 31

Augmented Rhombidodecadodecahedron facet (self intersecting) - R.Klitzing  click here for more on all three 

Augmented Conway 6-5-4 (coplanar faces) 

Augmented Conway 6-5-4 (no coplanar faces)

8-3-3-3 45 16, 32, 18 octagonal antiprism
8-3-4-3 15 28, 48, 22 augmented truncated cube (J66)
8-4-3-3 15 48, 116, 70 McNeill 8-4-3-3  click here for more
10-3-3-3 36 20, 40, 22 decagonal antiprism
10-3-4-3 6 65, 109, 42 augmented truncated dodecahedron (J68)
10-4-3-3 6 60, 145, 87 McNeill 10-4-3-3  click here for more

The only apolydronic tetrahedral acron to yield a solution is the N-3-3-3 which for all N yields the N-gonal anti-prism (e.g. 7-3-3-3).  Click here for a full list of potentialities.

Removing now our limitation to polydronic polygons, we can now consider the remaining apolydronic polygons.  Apart from the N-3-3-3 which for all N yields the N-gonal anti-prism (e.g. 7-3-3-3), acrons containing apolydronic polygons such as the heptagon seem to only rarely yield acrohedra.  The list below shows all apolydronic tetrahedral acrons known to date to yield solutions.  Click here for a full list of potential acrons.
 
trihedral
acron
Deficit (degrees) V, E, F Polyhedron
7-3-4-3 21 3/7 42, 82 ,41

44, 85, 42

McNeill 7-3-4-3 (coplanar faces and self-intersecting)

McNeill 7-3-4-3 (non-coplanar faces and self-intersecting)  click here for more

Click here for flat tetrahedral acrons.

Pentahedral acrons

Turning to the pentahedral acrons, with the same restrictions as above, the list is short.
 
pentahedral
acron
Deficit (Degrees) V, E, F Polyhedron
3-3-3-3-3 60 6, 10, 6 pentagonal pyramid (J02)
4-3-3-3-3 30 10, 26, 18 sphenomegacorona (J88)
5-3-3-3-3 12 30, 80, 52 pentagonal bifrustocingulum     click here for more

No apolydronic pentahedral acrons exist as the smallest candidate, the 7-3-3-3-3, has a negative angular deficit.

Click here for flat pentahedral acrons.

Hexahedral acrons

Only one hexahedral acron exists, the 3-3-3-3-3-3.  It is flat.

Credits

Many of the polyhedra referred to in this list have been identified by a search through the uniform polyhedra and the Johnson Solids.  It is my expectation that a number of the minimal solutions given above will be improved upon.  Some of the acrons without solutions have already generated some interesting near misses.

I am indebted to Professor John Conway of Princeton University for introducing me to the concept of acrohedra and for providing the list of trihedral acrons about which this page is based, Professor Conway's work is presented here with his permission.  Thanks also to Melinda Green (who initiated the original search for the 5-5-4 and kicked this whole thing off) and to all contributors.

Any errors and omissions on this page are mine.

Credits for individual acrohedra are as follows:

5-4-3  Professor John Conway and Dr Richard Klitzing
6-5-4  Professor John Conway and Dr Richard Klitzing
10-6-3  Professor John Conway, Alex Doskey and Dr Richard Klitzing

7-4-3  myself
7-6-4  myself
7-7-3  Mason Green
9-4-3  myself
11-4-3  myself

6-3-5-3  Professor John Conway
6-5-3-3  Alex Doskey

7-3-4-3  myself

8-4-3-3   myself
10-4-3-3   myself

5-3-3-3-3   Janella Large and Mick Ayrton