Heesch tiling

Information on Heesch tiling can be found here… https://en.wikipedia.org/wiki/Heesch%27s_problem

 

The examples I found, were drawn up in VectorEngineer Quick-Tools with the aid of a grid (regular hexagons connected point to point).  I used Jasc PaintShop Pro V.7 to make them look pretty.

HexGrid

 

Here are the shapes that I decided upon.  Numbers inside denote number of sides.

Heesch0

 

Heesch number 4

Heesch 4e

 

Heesch number 5

DS - Heesch 5d

 

Heesch number 5

Heesch5b

 

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Floret pentagonal tiling and dart

By dissecting the pentagon on the far right into three distinct shapes, three other pentagons can be produced, including the floret (second in from left).  All but one can tile the plane – but which?

Floret and dart2

 

Below are a few examples of how the floret (coloured) and smaller dart, might fit together.  All patterns should be able to extend further.

Pent dart-5

Pent dart-6

Pent dart-7

Pent dart-8

Pent dart-3

Pent dart-9

 

… and a couple more with small and large regular hexagons at the centre.

Pent dart-2Pent dart-4

 

Can this chaotic pattern keep going?

Floret and dar2a

 

Here’s a few extras that I dug up, with the smaller darts highlighted.  The last pattern displays no symmetry.Floret and dart3e

Floret and dart3b

Floret and dart3dFloret and dart3a

Floret and dart3c

Cairo-Prismatic tiling – part two

Continuing with the prismatic chains theme…

cairo-prismatic tiling 7

cairo-prismatic tiling 14

Propellers – tri-fold rotational patterns with the latter exhibiting a dividing line.  If you were to omit this line and rejoin the segments, would the pattern then become a mirror image of the first?

 

cairo-prismatic tiling 10

An interesting combination of links, chains and gears (aka Christmas trees).

 

cairo-prismatic tiling 16

At first glace, the inset image on the right, looks like a magnified version of the one on the left.  In fact, it has grown due to more shapes being added to the motif.

 

cairo-prismatic tiling 17

Fort – a particular favourite of mine.  By mirroring top, bottom, left and right, the pattern will produce a more detailed wood grain effect than the example above it.

 

cairo-prismatic tiling 13

Whirlpool – this pattern has no symmetry.  If the seven spiral gears were to continue to form a closed loop, it would leave a regular hexagon void at its centre.