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 XYZacron (or an 'XYZ') as being a vertex surrounded in sequence by an Xgon, Ygon and Zgon then we can define an XYZacrohedron as being a polyhedron containing at least one XYZacron. A cube for example can be regarded as a 444acrohedron.
The image above shows a 943 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 543 have brought to light lesserknown polyhedra like the Stewart G3 or have required a specific construction, such as Professor Conway's solution for the 654. Others, like the 554, have so far eluded solution.
In certain cases the minimum acrohedron is not clearcut. Cases such as the 555 yield a different solution (the Stewart G3) for the minimum number of vertices than for the minimum number of faces (the dodecahedron). In such cases both polyhedra are listed.
In other cases the minimum acrohedron contains flat acrons ("Flat") (eg a 633 acron) and/or coplanar faces ("CoP") (e.g. the 654), or is selfintersecting ("SInt") (e.g. the 1053). For cases such as this all the minimal solutions 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 
333  180  4, 6, 4  tetrahedron see also nn3 acrons 
433  150  5, 8, 5  square pyramid (J01) 
443  120  6, 9, 5  triangular prism see also nn3 acrons 
444  90  8, 12, 6  cube 
533  132  6, 10, 6  pentagonal pyramid (J02) 
543  102  10, 18, 9 (SInt)
10, 20, 12 
Small Ditrigonal Icosidodecahedron facet (self intersecting)  R.Klitzing click here for more these 
544  63  10, 15, 7  pentagonal prism 
553  84  9, 15, 8  tridiminished icosahedron (J63) see also nn3 acrons 
554  54  none known  view some near misses 
555  36  13, 24,
13
20, 30, 12 
Stewart G3 click here for more 
643  90  9, 15, 8  triangular cupola (J03) 
644  60  12, 18, 8  hexagonal prism 
653 NEW 
72  28, 67, 41 (Flat, CoP) 29, 70, 43 (CoP) 38, 96, 60 (SInt) 
Rayne
653 (ver 2) (flat) Rayne 653 (ver 2) (coplanar) Rayne 653 (ver 2) 
654  42  27,
42, 17 (SInt)
18, 36, 20 (CoP) 22, 48, 28 
Rhombidodecadodecahedron facet (self intersecting)  R.Klitzing click here for more on all three 
655 
24  21, 43, 24 (Flat, CoP) 25, 55, 32 (CoP) 31, 69, 40 (SInt) 
Augmented
Q4
(flat) Augmented Q4 (coplanar) Augmented Q4 
663  60  12, 18, 8  truncated tetrahedron see also nn3 acrons 
664  30  24, 36, 14  truncated octahedron 
665  12  60, 90, 32  truncated icosahedron 
843  75  12, 20, 10  square cupola (J04) 
844  45  16, 24, 10  octagonal prism 
853  57  none known  
854  27  none known  
855  9  none known  
863 NEW  45  19, 39, 22(Flat, CoP) 21, 46, 27 (CoP) 27, 64, 39 (SInt) 
Rayne
863 (ver 3) (flat) Rayne 863 (ver 3) (coplanar) Rayne 863 (ver 3) See also n63 acrohedra. 
864  15  48, 72, 26  truncated cuboctahedron 
883  30  24, 36, 14  truncated cube see also nn3 acrons 
1043  66  15, 25, 12  pentagonal cupola (J05) 
1044  36  20, 30, 12  decagonal prism 
1053  48  20, 32, 14 (SInt)
20, 35, 17 
icosidodecahedron facet (selfintersecting) 
1054  18  45, 75, 32  tridiminished rhombicosidodecahedron (J83) 
1063 UPDATED 
36  19, 39, 22 (Flat, CoP) 26, 52, 28 (CoP) 40, 70, 28 (SInt) 30, 62, 34 
Rayne
1063 (ver 4) (flat) Rayne 1063 (ver 4) (coplanar) See also n63 acrohedra. Small icosicosidodecahedron facet (self intersecting)  R.Klitzing Conway 1063 (noncoplanar faces) click here for more on the Klitzing and Conway polyhedra 
1064 UPDATED  6  37, 78,43 (Flat, CoP) 38, 81, 45 (CoP) 49, 113, 66 (SInt) 120, 180, 62 
Rayne
1064 (ver 1) (flat) Rayne 1064 (ver 1) (coplanar) Rayne 1064 (ver 1) (selfintersecting truncated icosidodecahedron 
1083  21  none known  
10103  12  60, 90, 32  truncated dodecahedron see also nn3 acrons 
Removing now our limitation to polydronic polygons, we can now consider the remaining apolydronic polygons. Acrons containing apolydronic polygons such as the heptagon also yield acrohedra. The list below shows all apolydronic trihedral acrons known to date to yield solutions. Acrons containing 7, 9 and 11gons are listed seperately, those valid for all n (including n>=12) are then listed below. Click here for a full list of potential acrons.
Trihedral Acron 
Deficit (Degrees) 
(V, E, F) 
Polyhedron 
743  81 3/7  20, 40 ,22  743 click here for more 
763 NEW  51 3/7 
40,
95, 57 (Flat, CoP) 41, 98, 59 (CoP) 54, 136, 84 (SInt) 
Rayne 763 ver 2 (flat) Rayne 763 ver 2 (coplanar) Rayne 763 ver 2 (self intersecting) See also n63 acrohedra. 
764 UPDATED 
21 3/7  27,
58, 33 (Flat, CoP) 28, 61, 35 (CoP) 37, 85, 50 (SInt) 
Rayne 764T (flat) Rayne 764T (coplanar) Rayne 764S (self intersecting) See also n64 acrohedra. Click here for info on another 764 acron. 
773  42 6/7  28, 49, 22  Mason Green's Small Supersemicupola (selfintersecting) click here for more, see also nn3 acrons 
943  70  26, 52, 28  943 
963 NEW  40 
52,
123, 73 (Flat, CoP) 53, 126, 75 (CoP) 70, 176, 108 (SInt) 
Rayne 963 ver 2 (flat) Rayne 963 ver 2 (coplanar) Rayne 963 ver 2 (self intersecting) See also n63 acrohedra. 
964 NEW  10 
35,
76, 43 (Flat, CoP) 36, 79, 45 (CoP) 47, 109, 64 (SInt) 
Rayne 964T (flat) Rayne 964T (coplanar) Rayne 964S (self intersecting) See also n64 acrohedra. 
1143  62 8/11  32, 64, 34  1143 
1163 NEW  32 8/11 
64,
151, 89 (Flat, CoP) 65, 154, 91 (CoP) 86, 216, 132 (SInt) 
Rayne
1163 ver 2 (flat) Rayne 1163 ver 2 (coplanar) Rayne 1163 ver 2 (self intersecting) See also n63 acrohedra. 
1164 NEW  2 8/11 
43,
94, 53 (Flat, CoP) 44, 97, 55 (CoP) 57, 133, 78 (SInt) 
Rayne 1164T (flat) Rayne 1164T (coplanar) Rayne 1164S (self intersecting) See also n64 acrohedra. 
1263 NEW

30 
30,
63, 35 (Flat, CoP) 36, 81, 47 (CoP) 47, 113, 68 (SInt) 
Rayne
1263 ver 3 (flat) Rayne 1263 ver 3 (coplanar) Rayne 1263 ver 3 (self intersecting) See also n63 acrohedra. 
n44 
2n, 3n, n+2 
Ngonal prism, valid for all n. 

n63 NEW (n even)  3n+2,
8n, 4n+3 (Flat, CoP) 3n+3, 8n+3, 4n+5 (CoP) 4n+4, 10n+8, 6n+6 (SInt) 
Rayne n63 (ver 2). See also n63 acrohedra. Valid for even n>=8 (n=4 and n=6 are degenerate) Examples: n=12, n=14 

n63 NEW (nodd)  6n2,14n3,
8n+1
(Flat,
CoP) 6n1, 14n, 8n+3 (CoP) 8n2, 20n4, 12n (SInt) 
Rayne n63 (ver 2)
See also n63 acrohedra. Valid for odd n>=5 (n=3 is degenerate) Example: n=13 
Note that the 863,
1063 and 1263 acrohedra are special cases where the capping
n/2gon
of edge length 2 can be subdvided in an more efficient manner than the
generic solution. There is also a special case for the 2463 but
in
this instance the subdivision of the large dodecagon requires so many
facets it is more complex than the generic case, for example the non coplanar case has
(V,E,F) = (127, 335, 210), compared to the generic
solution with (V,E,F) = (100, 248, 150).
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 trihedral acron is the nn3 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 nonreflexive and must not be crossed. Acrons such as the crossed 4343 occurring in the tetrahemihexahedron will not be considered here.
In considering the tetrahedral acrons we must differentiate between the XYZZ and the XZYZ, as the sequence of polygons is relevant.
A list of the
polydronic
tetrahedral acrons follows:
tetrahedral
acron 
Deficit (degrees)  V, E, F  Polyhedron 
3333  120  5, 8, 5  square pyramid (J01) 
4333  90  8, 16, 10  square antiprism 
4343  60  8, 14, 8  gyrobifastigium (J26) 
4433  60  7, 12, 7  elongated triangular pyramid (J07) 
4443  30  12, 20, 10  square cupola (J04) 
5333  72  9, 15, 8  tridiminished icosahedron (J63) 
5343  42  14, 26, 14  bilunabirotunda (J91) 
5353  24  14, 26, 14  bilunabirotunda (J91) 
5433  42  11, 19, 10  augmented pentagonal prism (J52) 
5434  12  15, 25, 12 
pentagonal
cupola (J05) 
5443  12  20, 38, 20  elongated bilunabirotunda 
5533  24  10, 18, 10  augmented tridiminished icosahedron (J64) 
6333  60  12, 24, 14  hexagonal antiprism 
6443  30  15, 27, 14  augmented truncated tetrahedron (J65) 
6353  12  38,
81, 45
40, 87, 49 
Conway 6353 (coplanar faces) click here for more 
6434  30  13, 22, 11  augmented hexagonal prism (J54) 
6533  12  28,
46,
20
19, 40, 23 23, 52, 31 
Augmented Rhombidodecadodecahedron facet (self intersecting)  R.Klitzing click here for more on all three 
8333  45  16, 32, 18  octagonal antiprism 
8343  15  28, 48, 22  augmented truncated cube (J66) 
8433  15  48, 116, 70  8433 click here for more 
10333  36  20, 40, 22  decagonal antiprism 
10343  6  65, 105, 42  augmented truncated dodecahedron (J68) 
10433  6  60, 145, 87  10433 click here for more 
Removing now our limitation
to polydronic polygons, we can now consider the remaining apolydronic
polygons.
Apart from the N333 which for all N yields the Ngonal
antiprism (e.g. 7333),
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.
tetrahedral acron 
Deficit (degrees)  V, E, F  Polyhedron 
7343  21 3/7  42, 82 ,41
44, 85, 42 
7343
(coplanar
faces
and
selfintersecting)
7343 (noncoplanar faces and selfintersecting) 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 
33333  60  6, 10, 6  pentagonal pyramid (J02) 
43333  30  10, 26, 18  sphenomegacorona (J88) 
53333  12  30, 80, 52  pentagonal bifrustocingulum click here for more 
No apolydronic pentahedral acrons exist as the smallest candidate, the 73333, has a negative angular deficit.
Click here for flat pentahedral acrons.
Hexahedral acrons
Only one hexahedral acron exists, the 333333. 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 the late 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 554 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:
543
Professor
John
Conway
and
Dr
Richard Klitzing
653
Rayne
H.
654
Professor John Conway and Dr
Richard Klitzing
665 Rayne H.
863
Rayne
H.
1063
Professor John Conway, Alex Doskey, Dr Richard Klitzing and Rayne H.
1064 Rayne H.
743 myself
763 Rayne H.
764 Rayne H.
773 Mason Green
943
myself
963 Rayne H.
964 Rayne H.
1143
myself
1163 Rayne H.
1164 Rayne H.
n63 Rayne H.
n64 Rayne H.
5443
Mick
Ayrton
6353
Professor John Conway
6533 Alex Doskey
7343 myself
8433
myself
10433
myself
53333
Janella
Large
and
Mick
Ayrton