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 |
| 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 |
| 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 |
| 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) |
| 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 |
| 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) |
| 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 |
| 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 |
| 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
