Trapezia- and rhombi-faced polyhedron models

I would like to thank Adrian Rossiter who wrote the language scripts in his ‘Antiprism’ program (http://www.antiprism.com/index.html) for all of the following polyhedron models that I constructed. I recorded a short video animation of each (listed at the end of this post) using ‘Debut Video Capture’ by NCH Software.

I made the first of these models a while ago using a ‘Jovo Construction Set’… http://www.jovo.com/en_jovo.html. It has eleven faces, comprising five regular hexagons and six trapezia and displays mirror symmetry. The shorter side of the trapezium is half the length of the longer sides. The model is not dissimilar to the regular pentagon and trapezium model linked below… https://hedraweb.wordpress.com/2016/05/02/in-the-beginning/.

More recently, I put together six other cardboard models that utilise a different type of trapezium using the same slot-together method as ‘Itsphun’ shapes (https://itsphun.com/products) but only two were verified by Adrian as being geometrically constructible polyhedra.

The first of these looks like a spinning top. It has twenty faces consisting of twelve golden ratio trapezia and eight equilateral triangles. My model (not shown) had a tendency to only rest on the six trapezium faces that surround the centre triangles. It looks like the two views, below, generated in ‘Antiprism’.

The second of the two models reminds me of a tombola drum. It has twelve faces made up of six trapezia and six rhombi. The model can stand upright on any of its faces.

Here is a clearer presentation of it using Adrian’s ‘Antiprism’ software.

Interestingly, this model has 3D space filling tessellation properties.

Near misses – Adrian confirmed that model ‘B’ was not possible. After further scrutiny of the other three, I noticed obvious flexing of the card, so would not have worked either. A little disappointing really, as they all display interesting symmetries and have both left and right handed versions. A) 12 trapezia/2 triangles, B) 12 trapezia/4 triangles, C) 12 trapezia/6 triangles and D) 12 trapezia/8 triangles.

Animated video clip links…

Eleven faced polyhedron – https://www.mediafire.com/file/9rha41jgcq85shk/hex_trap.mp4/file

Twenty faced polyhedron – https://www.mediafire.com/file/1fzg59pghes5vos/trap_tri_spinning_top_1a.mp4/file

Twelve faced polyhedron – https://www.mediafire.com/file/ljwv7p5wbjqusqq/trap_rhomb_tombola_model1a.mp4/file

Space filling example – https://www.mediafire.com/file/viooi0i4xtqvk4t/trap_rhomb_tombola_space_filler1a.mp4/file

Parallelogram and a half

I worked on this parallelogram some time ago. The only other example of it that I found, was this radial tiling… https://www.geogebra.org/m/mCYSQSeu

I constructed the parallelogram using twelve iamonds

Reptiles (https://en.wikipedia.org/wiki/Rep-tile) can be made up from the sequence 1, 4, 9, 16, 25, 36 etc.

A few interesting periodic tilings

The shape can also be put together to produce diamonds…

… which in turn can make hexagons of unlimited size and designs.

Radials can be compiled from 60° wedges.

Zigzag formations are unlimited.

Radials with various shapes at the core.

A few motifs exhibiting 3D illusion qualities.

… as well as chaotic tilings that continue forever.

Hedraweb listing

I thought it was time I listed all my posts since 2016 to make it easier to find a particular article.

All you have to do is left click the search icon (top right of page) and enter a keyword.

Penguin nonagon

I composed this shape (in 2019) by overlapping decagons, a technique that was employed by Robert Reid in the book ‘The Gentle Art of Filling Space’. However, my motif generating procedure differs, in that the decagon overlaps occur at midpoints between decagon vertices as opposed to Robert’s coincident vertices (thank you to Michael Dowle for pointing that out).

I later stumbled across a similar shape, discovered by Tim Lexen, that uses only three arcs, called a tricurve. Both the penguin and tricurve can tile in the same way but according to Tim, the penguin has more tiling possibilities.

http://paulbourke.net/geometry/tricurves/

https://aperiodical.com/?s=tricurves

I like the tricurve for its smoothness but I find the penguin better to work with, as it has points of reference, such as edge midpoints and vertices.

Penguins can be put together in a number of ways.

A typical radial tiling.

Two-fold rotation.

A random selection of central cores that can expand forever.

The beginnings of a single tailed zigzag spiral.

Modified central cores.

Tilings with holes. Small and large decagons and octagons.

An attractive decagram and five pointed star.

Just found this five tailed spiral. I’m quite sure there are others.

More polycairo tiling

Here is another shape I made up from eight polycairos dubbed the boar.

This new shape can be completely surrounded by 5, 6 or 7 copies of itself.

Central cores of four boars, can be constructed in two ways (light and dark grey). The resulting patterns can be both periodic or aperiodic. Coronas that fully surround the cores, overlap.

Both core types can be surrounded by all eight rotations and mirrors. Unlike the previous examples, the boars that make up the coronas, are not shared.

Boars can produce chains or strips of alternating rotations or mirrors. These can in turn, join adjacently to themselves or other combinations.

In the second of the two patterns below, there are diagonal rows of boars, left to right, which use all four rotations and their mirrors (two different sequences), running either side of two hidden cores.

A few more examples to finish…

A pattern with a kink
A strange mix
Four fold rotational symmetry
A rotational pattern around two central cores

Software used… Polyform Puzzle Solver by Jaap Scherphuis https://www.jaapsch.net/puzzles/polysolver.htm and a very old version of Jasc Paintshop Pro 7.0.

Polycairo tiling

I have lately been experimenting with shapes composed of Cairo pentagons, with the aid of the  Polyform Puzzle Solver by Jaap Scherphuis.  One of the most interesting of these, is the fish.  Alain Nicholas worked his magic on it, creating a piranha, soon to be shown on his amazing site… http://en.tessellations-nicolas.com/

Piranha

 

It can produce aperiodic tilings within a square border of various sizes.

1a-aperiodic with border

 

This example displays the frequency and placement of rotations and mirrors.

DS-Cairo10

 

Smaller motifs can be arranged in different ways.  The bottom example had me fooled at first – the two images are actually mirrors of each other.

Cairo motif examples small

 

Here’s another polycairo made up of 12 Cairo pentagons.  It is easier to tessellate than the fish, as you only have to deal with rotations, due to its mirror symmetry

Goat

 

… and here’s the artwork that Alain Nicholas created from it – thank you kind Sir!

Nicholas-billy goat

 

A chaotic arrangement of the tiles.

Goat1

 

I believe this last example can only tile periodically but in an interesting way.

Cairo-8 rots

 

The pattern uses all four rotations and their mirrors.

Cairo-8a rotations

 

Thank you to David Bailey for helping out.  His site on Escher-like tessellations and Cairo tiling is well worth a visit… http://www.tess-elation.co.uk/new-hom

 

 

 

 

 

 

 

 

 

Aperiodic hendecagon tilings

Drafter tiling piece

This concave prototile consists of six kites (twelve drafters).  I used the Polyform Puzzle Solver by Jaap Scherphuis to explore the tiling possibilities.  The applet is available here… https://www.jaapsch.net/puzzles/polysolver.htm  There are at least two periodic tilings (not shown).  Grid boundaries are user defined.

 

Whilst the silhouette (left) has three-fold symmetry, the hendecagon tile placements (right) interestingly display no symmetry.

 

Drafter tiling 10-12960

According to the solver program, the above example has 12960 solutions but this will of course include many duplicates (rotations etc).

 

Hendecagon seed

The three shapes highlighted in yellow, were manually placed.  The surrounding tiles were generated by the solver program.

 

Drafter tiling2

This smaller example demonstrates the aperiodic nature of the tiling.  Hendecagons can be continually placed ad lib, without having to backtrack.

 

Below are a few other shapes (n-kites) that I experimented with…

 

Drafter-9

Periodic zigzag pattern (6 kites)

 

Drafter-squirel

Tri-fold rotocentre (7 kites)

 

Drafter tiling01

Hexagonal rosettes (6 kites)

Shoveler tilings by Araki Yoshiaki

Araki has produced a short video based on my Shoveler tile.

https://drive.google.com/file/d/1r58sbAcLv0Lva5O_keKX4iNrvcIHUP5E/view

These following examples were sent to me by Araki Yoshiaki (www.tessellation.jp)

Yoshiaki Araki-2
Chaotic

araki_shoveler_radial
Three-fold rotocentre

Yoshiaki Araki-1
A non-trivial p3 periodic tiling

Below are links to my original shoveler articles.

https://hedraweb.wordpress.com/2017/04/08/the-shoveler/ https://hedraweb.wordpress.com/2017/04/13/shoveler-motifs/ https://hedraweb.wordpress.com/2017/05/01/shoveler-update/

Dicrowns and ‘the foal’ motif

Crowns are covered in the book ‘The Gentle Art of Filling Space’ published by Lulu books.  It is available as a portable document file and costs 15 Euros (excl. VAT).  There is also a paperback version.

Gentle Art cover

 

There are seven dicrowns, three that tile the plane (in turquoise) and four non-tilers (in grey) circled below.

Mule motif tiling7

 

One of the non-tilers below, is a special case (Heesch 1).  This dicrown needs fourteen copies of itself, for it to be completely surrounded.

For more on heesch tiling, please visit here… https://hedraweb.wordpress.com/2019/05/16/heesch-tiling/

Mule motif non tiler

dichrome heesch 1

 

The three turquoise dicrowns (circled above), tile the plane.  Two of which (on right) can be superimposed on many of Robert’s crown tessellations (diagrams 64-68, 73 and 74 in ‘The Gentle Art of Filling Space’).  The third one on the left, I have dubbed ‘the foal’ and is featured below.

Strips of these can be joined together indefinitely for periodic tilings.

Mule motif

Mule motif tiling2-a

Mule motif tiling-a

 

There are also radial patterns – most have no symmetry.  I  began with the 10 fold pattern on the left.  By mirroring groups of three ‘foals’, radiating from a common point/pivot, new structures/patterns appeared.  Red indicates a clockwise positioning of the ‘foals’ around the pivot and blue, counter clockwise.  Once a core pattern radiates, the structure of the tessellation is determined and cannot change.

Mule motif tiling4

 

By experimenting further, I achieved more elaborate starting positions.

Mule motif tiling3

This is the first example after a few iterations…

Mule motif tiling6

and another that contains no symmetry…

Mule motif tiling5