Lmst

Infinite grill

#AlgorithmicArt #CreativeCoding
#Processing #glsl #shaders #hexagons

Red, yellow, and blue paths over dark green hexagons branch and join into an infinite tiling of interwoven sheets.

The polyominoes with perimeter 10 have area 27, so three copies of them can make a 9×9 square. Peter Esser has a good page on tilings of multiple sets of polyominoes where each set is a different color, no polyominoes within the same set touch, and (ideally) no polyominoes of the same shape touch either. (See https://polyforms.eu/coloredpolyominoes/start.html) Isoperimetric polyominoes are discussed on the Poly Pages here: http://recmath.org/PolyPages/PolyPages/Isopolyos.html

I found this manually. A strict three coloring (one where no pieces of the same color meet at corners) would be preferred. I'm not optimistic about there being a solution where no polyominoes of the same shape touch.

#TilingTuesday

xくんが死んだ😔

#TilingTuesday #mathart #tiling #abstract #knot #art

Almost forgot a #tilingtuesday offering..

UltraFractal6 Wreno in OM2

$A tiled complex fractal pattern in warm brown, orange and yellow$

A stone wall with extruded mortar, Troutbeck, Cumbria, England

#TilingTuesday #geometry #tiling #MathArt #photography #design #pattern #architecture #stone

A wall made of rough grey/brown irregular stones. The mortar is proud of the stones and forms neat lines between the stones. Many of the stones are almost long rectangles; they are contrasted by shorter rectangles and other shapes like triangles, pentagons and a stone that looks like an aerofoil.

RE: https://mathstodon.xyz/@robinhouston/115531286196747333

#tilingTuesday

Hex with a twist

#TilingTuesday

#AlgorithmicArt #CreativeCoding
#Processing #glsl #shaders #hexagons

I think it would be appropriate to share some of my pictures of Ayudha Pooja art for #TilingTuesday

#tilingTuesday

A white spherical object (an antenna of some sort?) atop a trestle platform.

The surface of the sphere is divided into pentagonal and hexagonal panels, but not in the usual way. Each hexagon has two consecutive angles of _almost_ 90°, which then match up to a corresponding angle of _almost_ 180° on another hexagon. The other three angles are close to 120°.

The visible pentagons appear to be regular.

For #TilingTuesday, I'd like to introduce an infinite family of aperiodic Wang tile sets, two for each real algebraic quadratic integer that is not the square root of a natural number. This generalises Sébastien Labbé's metallic mean tiles (see https://arxiv.org/abs/2312.03652 and https://arxiv.org/abs/2403.03197), and the construction lends itself to a different proof of aperiodicity (basically "proof by Tangram" plus a bit of algebra) that avoids the need to prove a tiling hierarchy. If you're interested in the details, you can find them here:
https://drive.google.com/drive/folders/1AzhobFNKL0cGboXaJ7thnEl6SAvq8C3l?usp=sharing,
but I thought it would be nice to demonstrate the argument for a concrete example.

Consider the set of 30 tiles shown in the first image, which can be considered Wang tiles if we assign colours to edges depending on the colour and direction of the arrows. Notice that if one side is red or orange, so is the opposite side. Thus we can categorise the tiles into four groups, depending on whether the vertical edges are red/orange and whether the horizontal edges are red/orange.

Suppose there was a periodic tiling with these tiles. Then we could find a square repeating unit. Let $ n $ be the number of its rows. Each row must either have all vertical edges red/orange, or none. Likewise each column must have all horizontal edges red/orange, or none. Let $ x$ and $ y $ be the number of red/orange columns and rows, respectively. Then in the repeating unit, the numbers of tiles with all edges red/orange, only vertical red/orange edges, only horizontal red/orange edge, and no red/orange edges are $ xy $ , $ (n-x)y $ , $ x(n-y) $ and $(n-x)(n-y) $ respectively.

(1/n)

A set of 30 tiles arranged in 5 rows of 6, edges are coloured purple, blue, green, red and orange and have arrow markings at their midpoints. The set of tiles is invariant under rotation by 90 degrees, and tiles are arranged so that orbits of the rotation action are contiguous. The last two tiles, which have purple arrows going around clockwise, and orange arrows going around anticlockwise, are invariant under the rotation, but otherwise the orbits contain four tiles

A 16 by 16 patch of tiles using the prototiles in the previous image

A diagram consisting of six polygons, coloured either purple or yellow. Vertices are labelled using combinations of beta, beta subscript star, i (the imaginary unit), m, 0 and 1. These specify the positions of vertices on the complex plane. The polygons are arranged in two columns, and in each column, the superposition of the top and middle polygons produces the bottom polygon, if yellow and purple areas cancel out.

This image and the next are included here to give an indication of the flavour of the proof.

Similarly to the previous image, this shows yellow and purple polygons with labelled vertices. Here, the left and middle polygons both consist of two different-sized right-angled triangles whose hypotenuses have the same slope, and are placed edge to edge, sharing a single endpoint. The superposition of the left and middle polygons produces the polygon on the right, after cancelling overlapping purple and yellow areas.

There are 20 ways (up to rotation and reflection) to make polyiamonds where each cell edge has one or two notches, and the total number of notches on the perimeter is 7. These have total area of 81, and tile a triangle nicely. (There is an odd number of 2-notch edges, so they cannot all be internal.) These can be seen as an extension of isoperimetric polyiamonds where cell edges can have different weights. A complete set of perimeter-weight-7 polyiamonds would also contain shapes with edges of weight 3, 4, and 5, but this subset was easier to work with.

#TilingTuesday

Tiling ðe surface of a polycube wiθ 1 of each polyominoid wiθ 1-3 squares 🫪

#3d #360fov #tilingtuesday #mathart #geometry #tiling

Blender/Octane/Vectron-xenodreambuie colour fragment. for #tilingtuesday ... I think it tiles at a slight offset, remember the dark bits are just shadows!

$A tiled fractal landscape of ridges, towers, peaks and troughs, repeating in an offset grid structure, seen from above, stark black hard shadows define the highs and lows. The colours are pinks and purples.$

#tilingtuesday
Decagons, pentagons, stars & skinny rhombi
edit: better resolution

Irregular tiling made of four different tiles

Window grille, Stavanger, Norway

#TilingTuesday #geometry #tiling #MathArt #photography #design #pattern

A white window grille made of curved five-pointed stars like an asterisk *. Some stars meets three at a point, in other places there are gaps. The gaps have different shapes like stars, rhombuses and concave hexagons.

#TilingTuesday

#TilingTuesday Decomposition of decagon into pentagons and golden isosceles triangles.

There's a small cheat where some triangle sides meet two other polygons instead of just one. You could view these large triangles as composition of 2 Golden Isosceles and 2 Golden Gnomon and then any polygon side is only shared with one other polygon, but that just looks less neat.

#Geometry #MathArt #Maths #MathsArt #Tiling #Art