How to use Tiling Patterns & Symmetry

Two ways to move, and why there are two

Every other viewer in this family has one camera. Kaleido has two, and once you know that, everything else about the controls makes sense.

A one-minute first tour

If you would rather learn by trying things, start here:

Drag with one finger or the left mouse button. You are travelling across the tiling, not sliding a sheet of paper.
Pinch or scroll. The frame grows or shrinks, but your place in the tiling does not change.
Open the side menu and choose {7,3} on the Poincaré disk. This is the clearest starting view of a curved tiling.
Open the gear menu and try Tile outline, then an ornament. The world stays the same; only the design printed on it changes.
Drag the round turn handle. Release it near a snapping point and notice that the pattern lines up with itself again.
Double-tap or double-click empty space to return home.

The two menus answer different questions: the side menu asks "which world am I in?" and the gear menu asks "what is drawn on that world?"

Six useful words

A tile is one shape in the pattern. A tiling covers a surface with tiles without gaps or overlaps.
Regular means that all sides have equal length and all corners have equal angles, measured within the geometry being used.
A symmetry is a move — such as a slide, turn, or reflection — that leaves the complete pattern looking unchanged.
A motif is the artwork repeated by those symmetries.
A projection or model is one way to draw the curved world on a flat screen. Different projections preserve different features.
The horizon is the boundary shown around some projections. It is infinitely far away and is not an edge of the world.

The curved scenes show a world with no zoom. That sounds odd, so here is what it means. A hyperbolic plane has a built-in size — its heptagons are a particular size, the way a metre is a particular length. You can travel closer to any heptagon or move into its centre, but you cannot scale the world so that every distance becomes twice as large. Making the picture bigger is a separate act, like holding a magnifying glass over a map: it does not move you, it just enlarges what you are already looking at.

So the two controls are:

Travel — you move through the world. Tiles slide past you and new ones arrive from the horizon forever.
Frame — you enlarge or shift the picture of the world, magnifying glass style. Use it to inspect the crowd of tiles near the horizon without leaving your current location.

Travelling toward the rim never gets you there. The rim is infinitely far away. That is not the app running out of room; it is what infinity looks like when it has been drawn on a screen.

The flat scenes have no such distinction — there, one finger pans and two fingers zoom, exactly as you would expect.

An everyday comparison may help. Walking toward a poster changes your position; using a magnifying glass changes only the picture reaching your eye. Travel is the walk. Frame is the magnifying glass. They may look similar for a moment, but only travelling carries you to a new tile.

Touch

Travel: drag with one finger.
Frame: pinch with two fingers to magnify, or drag with two fingers to shift the picture.
Turn the pattern: drag the small round handle. See below.
Reset: double-tap an empty area to fly home.

Mouse and trackpad

Travel: click and drag.
Frame: scroll to magnify. To shift the picture, hold Shift and drag — or drag with the middle or right button. A mouse has no second finger, so these stand in for the two-finger gesture.
Turn the pattern: drag the round handle, or press [ and ].
Reset: double-click an empty area.

The turn handle

On the curved scenes there is a small round dot sitting on the pattern. It is not a decoration — it is a grip. Drag it around and the whole world turns beneath you.

It is stuck to one point of the tiling, like a pin pushed into the pattern, so when you travel it travels too. Pin it in a cell and it stays in that cell, in the same spot within it, for as long as you keep it. If you travel far enough that it leaves the screen, it parks at the nearest edge and grows a little arrow pointing to where it really is.

Turning snaps to the angles that map the tiling onto itself. On {7,3}, sevenths of a turn snap; on {5,4}, fifths. Let go near one of those and the pattern clicks back into place, which is a satisfying way to feel what "7-fold symmetry" actually means. [ and ] step by exactly one of those angles, so the keyboard gets there too.

Choosing what you are looking at

The side menu (the ☰ button, left) chooses the world: three flat kaleidoscopes, then the curved ones. The curved list has two things going on — several different tilings, and several different pictures of {7,3}. Both are worth trying. Each scene has its own help page under the ? menu explaining its tiling and, for the curved ones, the particular way its picture distorts.

The gear menu (right) chooses the pattern drawn into the tiles. Four are offered directly; More patterns holds the rest, including the line-only views that show you the tiling's skeleton. Custom is where your own drawings live.

Two patterns are only offered where the geometry allows them, and say so when it does not, because "there is no such colouring for this tiling" is a genuine fact about the shape rather than a missing feature:

Checkerboard needs an even number of tiles meeting at each corner. {5,4} has it; {7,3} cannot.
Panels — the football pattern — needs the truncated form, which every curved scene has and the flat ones do not yet.

The names in braces are counting instructions. In {7,3}, every tile has seven sides and three tiles meet at each corner. In {4,5}, the tiles are squares and five meet at each corner. Try counting with the line-only Tile outline pattern if an ornament makes the tile boundaries hard to see.

The names beginning with a star describe mirror symmetry. For example, *632 has three important rotation points: one that repeats after one-sixth of a turn, one after one-third, and one after one-half. You do not need to memorize these names to use the app; they are labels that let the help pages connect the picture to standard mathematical notation.

Drawing your own

Open Custom → New / manage… in the gear menu and you can draw straight onto the tiling; every mark is instantly repeated everywhere the symmetry says it should be. That has its own help page, Drawing your own motif, which is worth reading first — the surface you draw on is smaller and stranger than it looks.

Light, dark and full screen

The theme follows your system setting and can be switched in the menu. The full-screen button on the canvas gives you the pattern and nothing else, which is the best way to look at these.

If the view becomes confusing

Dragging moves the frame instead of the world: on touch, use one finger for travel and two for the frame. With a mouse, release Shift and use the left button.
A pattern is greyed out: pause on its explanation. Some colourings are mathematically impossible on the current tiling; choosing another scene can make them available.
The turn handle is at the edge: its pinned point is off-screen. The arrow shows where it is. Reset if you want to bring it home.
A custom drawing looks mirrored many times: that is the normal kaleidoscope mode. The Drawing your own motif page explains how to use the whole tile when you want an asymmetric design.

Sharing

The scene, the pattern, and exactly where you have travelled to all live in the address bar. Copy the link and whoever opens it arrives at the same place in the same world, looking the same way.

Three guided challenges

Once the controls feel natural, try these:

Find the horizon. In the Poincaré disk, travel toward the rim for a while. Watch tiny tiles grow as they approach the centre while new tiny tiles take their place. The rim never gets closer.
Compare two maps. Keep the {7,3} tiling and switch between Poincaré and Klein. Which view preserves the shape of a corner? Which makes tile edges look straight?
Test a colouring. Try Checkerboard on {5,4} and {7,3}. Count the tiles around a corner and explain why alternating two colours succeeds only when that count is even.