n-4-3-3 polyhedra

8-4-3-3 (left) and 10-4-3-3 (right)

The solutions to the 8-4-3-3 and 10-4-3-3 acrons are obviously related.  They are members of a family of polyhedra formed by wrapping a complex of squares and triangles between two polygons.   The squares are in strips of three, the triangles are in complexes of fourteen.

In addition to the above linked octagonal and decagonal models, examples can be formed for other n-gons, where n is even.    The figures continue above n=12,but as there is now an angular excess the triangular complexes become folded.

n-4-3-3 with n odd

'7'-4-3-3

Due to the alternating nature of the connecting squares and triangles, examples cannot be formed where n is odd.  However a closely related toroidal family can be generated by utilising the degenerate 2n/2-gons (such as the 14/2-gon).  Examples generated using these polygons can then have the base polygons removed leaving the edges of the square and triangular rings connected.    The resulting toroid has an n-gonal prismatic symmetry.  A number of examples are included in the zip files below.

Again, models can be generated for n/d where d>1:

'9/2'-4-3-3

Again a number of examples are included in the zip files below.

Isomerism

There are a number of distinct isomers for each n - although some appear to have limits as shown below.
The names are informal and are intended solely as an aid to recognition.
It is not known if the list below is complete.

The zip files linked below each model contain WRL, OFF and TXT files for each model in the range.  Where they can be generated, models are included for n = 5/2, 3, 7/2, 4, 9/2, 5, 11/2, 6, 7, 8, 9, 10, 11, 12, 14, 16.
The links in the above list are zip files of each isomer for that value of n.   A large zip file containing both sets of zip files is here.
Note that in the isomer specific zip files below the star polygons have been renamed, for example, 5/2 is renamed 2d5 (2 decimal 5).  This is so the models are presented in strict sequence of increasing X.

The discovery, generation and classification of these isomers is a joint effort in October 2020 with Roger Kaufman, the forms have been generated using a combination of HEDRON, Stella and Antiprism.

Type1. Prismatic Symmetry

 6-433_exo   WRL  OFF n-433_exo.zip (n = 5/2 - 16) 6-433_endo   WRL  OFF n-433_endo.zip (n = 5 - 11/2) 6-433_inverted exo   WRL  OFF n-433_inverted_exo.zip (n = 5/2 - 16) 6-433 endo crossed   WRL  OFF n-433_endo_crossed.zip (n = 4 - 16) 6-433 double crown   WRL  OFF n-433_double_crown.zip (n = 4 - 16) 6-433 folded   WRL  OFF n-433_folded.zip (n = 5/2 - 14) 6-433 saucer   WRL  OFF n-433_saucer.zip (n = 5/2 - 16) 6-433 saucer2   WRL  OFF n-433_saucer2.zip (n = 5/2 - 14) 6-433 saucer3   WRL  OFF n-433_saucer3.zip (n = 5 - 16) 6-433 exo folded   WRL  OFF n-433_exo_folded.zip (n = 5/2 - 16)

Type2. Chiral Prismatic Symmetry

 6-433_chiral   WRL  OFF n-433_chiral.zip (n = 5/2 - 16) 6-433 inverted exo chiral   WRL  OFF n-433_inverted_exo_chiral.zip (n = 5/2 - 16) 6-433 chiral folded   WRL  OFF n-433_chiral_folded.zip (n = 5/2 - 12) 6-433 chiral saucer   WRL  OFF n-433_chiral_saucer.zip (n = 4 - 16) 6-433 exo chiral folded   WRL  OFF n-433_exo_chiral_folded.zip (n = 5/2 - 16)

Type 3: Pyramidical Symmetry

 6-433 cup   WRL  OFF n-433_cup.zip (n = 5/2 - 16) see also coplanar cases below 6-433 crowned hatbox   WRL  OFF n-433_crowned_hatbox.zip (n = 5/2 - 14) 6-433 lighthouse   WRL  OFF n-433_lighthouse.zip (n = 4 - 16) 6-433 crown   WRL  OFF n-433_crown.zip (n = 9/2 - 16)  4 is degenerate 6-433 crown2   WRL  OFF n-433_crown2.zip (n = 9/2 - 12) 6-433 crown3   WRL  OFF n-433_crown3.zip (n = 3 - 10) 6-433 hatbox   WRL  OFF n-433_hatbox.zip (n = 5/2 - 16) 6-433 crossbox   WRL  OFF n-433_crossbox.zip (n = 5/2 - 11) 6-433 crossbox2   WRL  OFF n-433_crossbox2.zip (n = 4 - 14) 6-433 eggtimer   WRL  OFF n-433_eggtimer.zip (n = 5/2 - 16) 6-433 endo cup   WRL  OFF n-433_endo_cup.zip (n = 3 - 11)

Coplanar cases

A number of coplanar cases can be formed.  Each triangular complex can be divided into two blocks of 3 triangles adjoining the bases and the central block of 8 triangles.  Any number of these central blocks can be everted or inverted.  One or both primatic caps (the prismatic base along with it's adjoining sides) can also be inverted or everted.  For the 6-433 this makes a total of 12 cases.  WRL and OFF files for all 12 cases are in this file:  6-433_Coplanar.zip

Further unexpected coplanar cases were also discovered by
Roger Kaufman. Each has only one coplanar prismatic base, in all cases this base can be everted or inverted - the images are the everted versions with the coplanar base at the bottom:

 6-433 'coplanar cup' Everted  WRL  OFF 6-433 'coplanar cup' Inverted  WRL  OFF The everted form is remarkably similar to 'cup' but they are two distinct forms. 6-433 'coplanar1' Everted  WRL  OFF 6-433 'coplanar1' Inverted  WRL  OFF 6-433 'coplanar2' Everted  WRL  OFF 6-433 'coplanar2' Inverted  WRL  OFF 6-433 'coplanar3' Everted  WRL  OFF 6-433 'coplanar3' Inverted  WRL  OFF

Coplanar cases have not been explored for n>6.

n-3-3-3-3 acrohedra.

4-3-3-3-3

A further related family exists where the square faces have been omitted and the triangular complexes attached to each edge of a base n-gon.    The toroidal family above has no equivalent.  Once more the above isomers are apparent,   models are the 'exo' isomer unless noted:
(Again thank-you to
Roger Kaufman for a number of the models below)

 WRL OFF exo 3  4  5  6  7 3  4  5  6  7 endo 8  9  10  11  12  14  15  16  17  18 8  9  10  11  12  14  15  16  17  18

Note: The n=6
model is not as accurate as normal as the near coplanar faces make rendering difficult.

Star ploygons are once again possible

7/2-3-3-3-3