Regular hexagon and square tiling

The excellent ‘Tyler’ app by Melinda Green and Don hatch, lets you explore planar tilings using regular polygons (from 3 to 12 sides). The zoom facility is fantastic. Get it here… https://superliminal.com/geometry/

There are some really good examples, as shown in the gallery… https://superliminal.com/geometry/tyler/gallery/ but in this instance, I am using only hexagons and squares.

To make it more interesting, I put a constraint on the placement of both hexagons and squares, i.e., no hexagon may touch another (except at vertices) and likewise for squares. As well as this, the gaps must be consistent; in this case, thin rhombi.

Below left shows the only possible dodecahedron configuration of hexagons and squares excluding rotations. Also, the ‘propeller’ to the right of it, will always appear in any pattern.

If we consider only squares when continuing a pattern from a dodecahedron at its centre, there arises many combinations.

In the example below, the ‘disk’, far right, is not permissible, as a hexagon within the dodecahedron will touch another on the outer rim.

When extending each of the twenty five arrangements a little further, ‘bites’ sometimes occur. These can be filled in three different ways.

Below – six examples of central cores other than the dodecagon arrangement.

I finish with an attractive aperiodic tessellation.

There is also another article here on these type of formations but without the constraints that I set. http://followinglearning.blogspot.com/2015/07/tiling-again.html

Joshua Socolar discovered a substitution tiling rule for this set of shapes… https://tilings.math.uni-bielefeld.de/substitution/socolar/

Penguin nonagon 2

It turns out that the ‘penguin’ nonagon can be constructed using any regular polygons, except pentagons and hexagons. I have ignored triangles and squares. You can view the original post here… https://hedraweb.wordpress.com/2021/04/02/penguin-nonagon/

So to recap, overlaps of polygons occur at midpoints between vertices. Only three polygons are needed but the visuals look quite attractive so I left them in. It is also worth noting that all other shapes produced within the regular polygon surrounds (except centres with odd numbers), can also tile but in less interesting ways.




The penguin will always be a nonagon but internal angles will change. The original penguin from overlapping decagons (below right) has identical internal angles for both feet (the only occurrence).

Because the pentadecagon is a factor of five, pentagon and decagon radials can still be achieved if you include central voids.

Here is one example with a pentagonal shaped void inner.

In the the examples below, I have used penguin from heptagons. I believe that concentric circles/loops, spirals and central voids will work in the same way with all regular polygons. However penguins from odd numbered polygons do not allow mirrored patterns.

Here I have produced two and three small spiral designs from the same penguin. Alas, a ‘galaxy’ spiral eluded me.

Aperiodic puzzle set

This imaginative set of shapes, is an adaption of the Penrose kites and darts and was thought up by Goossen Karssenberg. More info here… https://kaasfabriek.nl/projecten/puzzels-aperiodieke-betegelingen.

I used a sillhouette paper cutting machine to cut out the shapes using thin card. Goossen named these ‘heart’, ‘sun’ and ‘cloud’.

Of the 7 possible vertex neighbourhoods in a tiling of kites and darts, three configurations of the new set have ‘suns’ at the centre. These make up the Sun, Deuce and Jack.

Then we have Ace, Queen and King

And finally the Star which is the 5 dart equivalent.

The cardboard shapes were difficult to work with but I managed a couple of small decahedron tilings.

It’s worth noting that the ‘heart’, ‘sun’ and ‘cloud’ do not need any colour matching rules or notches to produce only aperiodic tilings.

Sets of these can be purchased in both wood and acrylic. Go here for sizing and prices.. https://www.goossenkarssenberg.nl/geometrische-patronen-geometric-patterns/puzzle-mosaics/

Mandelbulb 3D

This amazing 3D fractal program has been around for some time. I had used it before but only briefly, as I was unable to produce anything of interest. You can find it here… https://www.mandelbulb.com/

Only just recently, my son Louis re-introduced me to it and helped me create my first renders (below). The image on the right is where ambient lighting has been added to create depth.

There’s a Navigator window where you can move in and out of fractals, pan, rotate, and other controls that I have yet to fully understand. I later stumbled across the ‘MutGen’ button (mutation generator). This is where you can adjust a few sliders that will randomise parameters, swap formulas etc.

Mandelbulb 3D is very addictive and I have since produced over 400 parameter files that I can render at any size (there’s probably a limit though); a few examples below. The results resemble futuristic buildings, uncanny worlds, impossible structures and space wrecks.

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.