About Tiling Patterns & Symmetry

Infinite Tilings — flat and curved tilings on one canvas

Infinite Tilings (ιKaleido) is a kaleidoscope you can walk around inside. It draws repeating patterns the way a kaleidoscope makes them — one shape, reflected in mirrors, forever — and lets you travel through the result, change the mirrors, and draw your own shape into them.

Half the app is the flat patterns you already know from tiled floors and wallpaper. The other half is the curved worlds where the rules are different: places where seven-sided tiles meet three to a corner, or five squares fit around a point, neither of which can happen on any flat surface anywhere.

You do not need to know advanced geometry before you begin. If you can count sides, add angles, and compare patterns, you can investigate the main ideas. The notation in scene names is explained when it appears, and the pictures are meant to be tested by dragging, turning, and drawing rather than accepted on faith.

What you can explore

• Flat kaleidoscopes — three of the seventeen symmetry groups that any repeating flat pattern must belong to, from the plainest rectangle of mirrors to the richest symmetry a flat plane allows.
• Curved tilings — regular {p,q} tilings of the hyperbolic plane: heptagons three to a corner, pentagons four, squares five, octagons three, and more.
• Six different pictures of the same curved world — the Poincaré disk, the Klein disk, the half-plane, the band, the square and one fitted to your window. Each is a complete and honest picture; each distorts something different. Switch between them and watch the same tiling re-flow.
• A gallery of patterns to fill the tiles with, from ornamental rosettes to the bare line-work that shows a tiling's skeleton — including a checkerboard and a football pattern, each offered only on the tilings whose geometry permits it.
• Your own drawings. Draw one stroke and it appears everywhere the symmetry says it should, instantly, as you draw. Save your motifs and carry them between worlds.

Travelling, not zooming

The curved scenes have no zoom, because a curved world has a built-in size — its heptagons are a particular size. You can travel closer to one, but you cannot scale every distance in the world at once. So the app separates two things other viewers often run together: travelling through the world and framing its picture. One finger travels; two fingers frame. Travel toward the horizon and you never arrive, because the horizon is infinitely far away and there are always more tiles.

Every pixel worked out from scratch

There is no list of polygons anywhere in this app. For every pixel on your screen, it follows the pattern's mirrors backwards to find which part of the single master tile lands there, and draws that. Every pixel answers independently.

That is why the app does not depend on a large saved patch that eventually runs out, and why it can keep producing tiles toward the horizon. The mathematical rule is unbounded, although the computer still has the usual limits of finite-precision numbers.

Learning as you go

Every scene has its own page in the help menu explaining what its tiling is, why it cannot exist on a flat plane if it cannot, and how its particular picture distorts the world. There is a longer essay on where tilings came from — from the Alhambra to Escher to the aperiodic monotile found in 2023. It introduces specialist terms as they appear and uses only high-school algebra and geometry.

A good place to begin

Start with the flat hexagonal kaleidoscope and choose a line-only pattern so that its mirrors are easy to see. Then open {7,3} in the Poincaré disk. Count seven sides on one tile and three tiles at one corner. The first scene shows the most symmetry possible on a flat plane; the second shows the smallest famous step into hyperbolic geometry.

After that, keep {7,3} and change only the projection. You will see the same world drawn six ways. Finally, create a custom motif with one short stroke and watch the symmetry turn it into a complete design.

Who it is for

• Students can use the app to make angle sums, transformations, symmetry, and curved geometry visible.
• Teachers can compare models live instead of relying on separate static diagrams.
• Artists and designers can treat the mathematics as a pattern-making tool without first learning the formal theory.
• Curious explorers can simply travel, draw, and ask why the world behaves as it does.

The help menu supports both approaches: open How to Explore Interactively for the controls, Drawing Your Own Motif for the art tools, the current scene page for the mathematics in front of you, or A Short History of Tilings for the larger story.