Tilings and tessellations are such an interesting area of mathematics. They seem so simple that everything should have been discovered already, and yet new fundamental questions are raised every year!

In this case, researchers wondered about "soft" tilings, i.e. tilings with as few corners as possible and are entirely "rounded" elsewhere. It turns out they are everywhere in nature, both in 2D and 3D. And you can have tilings of three-dimensional spaces that have no corners at all!

https://www.reviewertoo.com/tessellation-revelation/
#TilingTuesday

Tilings and tessellations are such an interesting area of mathematics. They seem so simple that everything should have been discovered already, and yet new fundamental questions are raised every year!

In this case, researchers wondered about "soft" tilings, i.e. tilings with as few corners as possible and are entirely "rounded" elsewhere. It turns out they are everywhere in nature, both in 2D and 3D. And you can have tilings of three-dimensional spaces that have no corners at all!

https://www.reviewertoo.com/tessellation-revelation/
#TilingTuesday

Zellij floor tiles, Bahia Palace, Marrakesh, Morocco.

#TilingTuesday #geometry #tiling #MathArt #photography #design #IslamicPattern #IslamicArt

Zellij floor tiles, Bahia Palace, Marrakesh, Morocco.

#TilingTuesday #geometry #tiling #MathArt #photography #design #IslamicPattern #IslamicArt

Fragmentarium2.5 based on
Hyperbolic-Honeycombs by neozhaoliang
#tilingtuesday

Fragmentarium2.5 based on
Hyperbolic-Honeycombs by neozhaoliang
#tilingtuesday

Advent of Tilings - Day 23.1

Almost at the end we see the light through diamond and hexagon shaped tunnels of an infinite surface, again monohedrally tiled.

#math #geometry #3d #combinatorics #tiling #AdventOfTilings #TilingTuesday

Advent of Tilings - Day 23.1

Almost at the end we see the light through diamond and hexagon shaped tunnels of an infinite surface, again monohedrally tiled.

#math #geometry #3d #combinatorics #tiling #AdventOfTilings #TilingTuesday

Meet Taubin smoothing. It is a nice smoothing method that is great for smoothing voxel based meshes. So for smoothing stepped "Lego-like" meshes.

The left here shows a raw voxelised (thresholded) voxel boundary surface. On the right is the Taubin smoothed result.

Taubin smoothing works by first doing an aggressive Laplacian smoothing step (a bit like a blurring), which removes bumps etc, but which unfortunately can distort the shape too (e.g. unwanted shrinkage/growth). Next it uses an "inverse smooth" (a bit like a sharpening operation) to attempt to push back the shape a bit to avoid distortion. With the right parameter choices it can perform really well.

This animation is for a thresholded 3D image. The colors are the original voxel intensities. The resulting shape presents as a central sphere like blob contained in a dodecahedron "cage".

See also: https://doi.org/10.1145/218380.218473

#GeometryProcessing #JuliaLang #Makie #Meshing #TilingTuesday

Advent of Tilings - Day 17.1

A heftier caliber construction of 2304 tiles forming a genus 289 surface, intended for a sturdier branch. This would also have been a worthy contribution to #TilingTuesday which unfortunately was forgotten.

#math #geometry #3d #combinatorics #tiling #AdventOfTilings

Meet Taubin smoothing. It is a nice smoothing method that is great for smoothing voxel based meshes. So for smoothing stepped "Lego-like" meshes.

The left here shows a raw voxelised (thresholded) voxel boundary surface. On the right is the Taubin smoothed result.

Taubin smoothing works by first doing an aggressive Laplacian smoothing step (a bit like a blurring), which removes bumps etc, but which unfortunately can distort the shape too (e.g. unwanted shrinkage/growth). Next it uses an "inverse smooth" (a bit like a sharpening operation) to attempt to push back the shape a bit to avoid distortion. With the right parameter choices it can perform really well.

This animation is for a thresholded 3D image. The colors are the original voxel intensities. The resulting shape presents as a central sphere like blob contained in a dodecahedron "cage".

See also: https://doi.org/10.1145/218380.218473

#GeometryProcessing #JuliaLang #Makie #Meshing #TilingTuesday

New maths blog post! The fifth in my series about handling aperiodic tilings combinatorially with string-processing algorithms:

https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperiodic-refine/

In this instalment I've automated the process of converting a tiling substitution system into one for which you can build a deterministic transducer, if the starting system didn't already admit one. I discuss the algorithm, its limitations, and a particular success in which it found a more economical substitution system for the hat tiling than any I already knew of.

#TilingTuesday

Animated Wythoff tiling construction on a hexagonal grid on an infinite ribbon.

made with #p5js

#creativecoding #TilingTuesday #mathart #geometry #mastoart

Animated Wythoff tiling construction on a hexagonal grid on an infinite ribbon.

made with #p5js

#creativecoding #TilingTuesday #mathart #geometry #mastoart

Autumnal tiling for #TilingTuesday

#tiling #mathart

@globalmuseum Most appropriate for a #TilingTuesday.

Best effort embedding of an octagonal tiling in 𝑅³ of a surface of genus 111. Each octagon has vertex valences [3,4,3,3,3,4,4,4].

#TilingTuesday #math #geometry #ball

Best effort embedding of an octagonal tiling in 𝑅³ of a surface of genus 111. Each octagon has vertex valences [3,4,3,3,3,4,4,4].

#TilingTuesday #math #geometry #ball

Autumnal tiling for #TilingTuesday

#tiling #mathart