Bamboo shoot (heptagon) – Update

I first came across this 7-sided polygon here…

Eight of these fit into an 18-gon (octadecagon).

Heptagon and Hex0


I replaced the two heptagons in the centre (and any space left over) with six ‘thorns’ (below).

Heptagon and Hex7


There are now known to be 34 different arrangements of 6 ‘Bamboo Shoots’ and 6 ‘Thorns’ inside an 18-gon.  There are an additional 26 if the mirror images of those arrangements without mirror symmetry are included, making a grand total of 60 (that’s ten more than I found)!  The arrangements with grey colouring have mirror symmetry.  Thank you to Michael Dowle for supplying this updated information and new diagram.  The colours really help in understanding the method he used.

MDowle - Heptagon & Hex2


It seems that there are only three ways to tile the plane with the bamboo shoot and thorn, all creating ‘pulse’ tilings.  The second example is a mirror of the first.  Curiously though, mirroring the third is a duplicate of itself!?

Heptagon and Hex5Heptagon and Hex4Heptagon and Hex6

You are welcome to correct me on any errors.




Not so Square

I divided a square into two parts producing a kite and dart and includes angles of 30, 60, 90, 150 and 210.  The small equilateral triangle (below right) is used later on as a centre piece.

Square kite and dart-0


The kite and dart can be arranged in a certain way, to create an infinite tiling, radiating from a single point.  Zig-zags are also possible.

Square kite and dart-2Square kite and dart-3

Six fold rotational symmetry works using six ‘V’ tilings.

Square kite and dart-1


By combining groups of 60 and 90 degree angled sub tilings, larger patterns evolve.  Bright yellow areas are basically just squares (a kite and dart for each).


‘V’ tilings can also overlap (white dart in centre below).

Square kite and dart-8


Other permutations used solely or in combination, are also possible.  The fourth example below, is a mix from two out of the three patterns above it.

Square kite and dart-6


I tried creating a triangle but it left a hole.  However, by filling this with a small equilateral triangle (in blue), a pattern can still emanate from it, without any triangle repetition.

Square kite and dart-7


Wavy kite and dart revisited

Continuing on from the previous post, Michael Dowle made up some drawings of his own and has provided explanations to help illustrate the chirality property of the wavy kite and darts.

Michael Dowle-wavy kites and darts2

On left, my tessellation and far right, the same turned through 180° and mirrored.

The example in the middle, is drawn using a mixture of the two enantiomers of the blue wavy kites and only one of the two enantiomeric darts.  But you cannot mix these enantiomers, since their placement/relative positioning, now breaks Penrose’s rules (not easy to spot – ed).

Michael Dowle-wavy kites and darts1

The illustration above (5-fold rotational symmetry), uses 5 copies of my tessellation with some extra kites and darts to fill spaces.

Credit – Michael Dowle.

12-gon tiling

Robert Reid created many beautiful tilings by overlapping n-gons.  I tried this using three 16-gons which produced a 12-gon crescent.  I believe the resulting shape can tile forever in various guises.



The radial tilings (and spiral) below, are all different.  Three with imbalanced cores.



Witch’s hat dissection

The Witch’s hat (far left) cannot tile the plane.  But, by splitting it into two shapes (a house and one other), various new tilings can evolve.Witch's hat

All the drawings below were kindly sent to me by Michael Dowle.  They are all based on the house and cookie fortune example (second in from the left).  I edited the garish colours a little!Witch's hat dissection edit-1 - MD & DS

Michael stated that the radial tilings, proved to be quite challenging.  The first and second examples below, start off with the same core (the first was abandoned).Witch's hat dissection edit-2 - MD & DS