I was going through some stuff around Christmas time and found some really old dot matrix prints. I think I had a Atari 520 STFM and a Lexmark printer at the time. The first is a Heighway dragon curve discovered by John Heighway in 1966.

I’m not familiar with the other two but they look like they are based on six sided stars. They are interesting to study, as they are both made up of a single line from start to finish. Very decorative.

The red squirrel can produce an attractive periodic tessellation. It is made up of a large motif containing all six rotations and their mirrors.

Six grey squirrels that rotate around a common point, produce a polyhex (a regular heptahex). These motifs, whether mirrored or not, can abut one another in endless arrangements. The greyed out areas denote reflections.

Also, polyhex motifs can be aligned differently around a central polyhex, which will alter the overall appearance.

Here’s another pattern that can only be put together one way (two, if you include its mirrored counterpart).

There are various ways to string tiles together to form three types of ‘slip planes’. These can be used as building blocks in some situations. The first example (zigzag) includes reflections.

Wavy (low profile). I have found six unique configurations (reflections not shown).

Wavy (high profile). Seven unique configurations (reflections not shown).

Central cores change when combining different configurations of slip-planes.

Grouping of decagons can produce ever expanding ‘motifs’, which can produce some interesting formations.

A single decagon can completely surround itself in various different ways.

The group of decagons on the left produce a ‘sunflower’. A maximum of five adjacent ‘petals’ can be used in a pattern. Two groups of four are also possible.

The thick black line or ‘handlebar’ indicates where a choice can be made. For example, by positioning a decagon (third from the left) another ‘handlebar’ forms. Placing of single decagons thereafter, are more likely to become forced.

The decagons at the heart of this pattern display mirror symmetry. All other decagons placed outside of this are forced.

This impressive double pentagon star tessellation utilises all five decagon rotations and their mirrors.

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/

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.

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.

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.

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.

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.