Rhombiverse User Guide

Rhombiverse and its twin, Polyhedraverse, are two ways of looking at the same geometry. Rhombiverse is the landscape: the lattices themselves, stretching out in every direction. Polyhedraverse is the portrait gallery: the shapes that live in those lattices, one at a time, up close.

Here, every piece fills space perfectly on a real crystal lattice, so you can only put a piece where the lattice has room for it. Tap to add a piece, long-press to remove one, and look at what you've built in different views, from 1D up to 6D.

The first part of this guide walks through common tasks. The second part lists every control.

Getting started

Pick a dimension

After ENTER, the dimension picker opens: a slowly turning shape whose faces are 1D+, 2D, 3D, 4D, 5D and 6D, each marked with an icon. Hover over a face (or press and hold it on a touchscreen) to see its name, then tap the one you want. Drag to turn the shape and bring other faces round.

You can switch later from Menu → Change Dimension, or from the Wizard (top left), which lists every lattice in each dimension with its pieces as rotating wireframes.

Place your first piece

An empty world shows a cyan outline where the first piece goes. Tap it. Then tap a face of any piece (a side, in 2D) to add a neighbour on the other side of it.

You build one piece at a time. 3D starts with the RD (rhombic dodecahedron) selected. The piece you're placing is shown in the Shape button at the bottom left; tap it to choose a different one.

Remove a piece

• Phone or tablet: long-press the piece.
• Mouse: right-click the piece. Right-click removes in every mode.

To take back your last change, tap Undo (↶, bottom right). Hold it to scrub back several steps at once. Each dimension keeps its own undo history, so undoing in 2D never touches your 3D or 4D build.

Move the camera

• Rotate: drag with one finger, or drag with the left mouse button.
• Zoom: pinch, or use the scroll wheel.
• Pan: drag with two fingers.

Choosing what to build

The pieces, by lattice

1D+ Signal (Wizard → 1D+ → Signal): a single line of bullet-shaped cells carrying a message. Type a message on the bottom line (it replaces the one before; edit it as usual): it appears as ghost cells, its Morse code, no need to know it, with the next one in orange. Tap it to place one at a time, or press Send (or Enter) to place the rest and set the message moving along the line, away from you (■ stops it). With the field empty, it shows what the line says. The eye button switches between Outside (from behind and above: the line runs up the screen and narrows into the distance, a faint mist carrying it on toward infinity) and Inside (in the tunnel: the cells pass just beneath you and climb away to a point in the distance). Long-press removes a cell and the line closes up behind it: it's one-dimensional.

The round pulse key beside the message is a telegraph key: tap for a dot, hold for a dash; a short pause starts a new letter and a longer one a new word. What you key shows in the message field as dots and dashes (· –), a space between letters and / between words; edit them there like any text (typing . and - works too). The little scroll beside Send opens the Morse code glossary: every letter, digit and sign with its dots and dashes; tap one to add it to your message. While the signal plays, the message reads out at the top of the screen, letter by letter, as it arrives there. Inside, drag to look around, pinch (or scroll) to move along the tunnel, and drag two fingers up or down (or Shift + scroll) to rise or sink; let go and the view eases back. The key learns your own rhythm, so key as slowly and carefully as you like: it reads your short presses as dots and your long ones as dashes, and judges your pauses against each other. The letters it reads show above the message box as you key.

1D+ Construct (Wizard → 1D+ → Construct · Square): build a square from 1D cells, one tap at a time, one line at a time. Only the line you're on shows; the next cell is orange: tap to fill it. The first side runs up the screen along X; at the corner a junction glows, Y joins (X stays) and the line you've built ghosts out; carry on clockwise: along the top, down the far side (a second X line, parallel to the first) and back along the bottom. When the loop closes, the whole square shows and fills in, and Open in 2D takes it to 2D's Square tile. Long-press takes back the last cell.

Then the cube: when the square closes, the view turns to three-quarters and Z rises from the start corner. Build its first edge one cell per tap; after that, each tap fills a whole edge: the other three Z edges, then the top square. Every edge carries its numbered name (X1, Y2, Z3 …); finished edges ghost, and the one just finished stays solid until the next. When the cube closes it shows whole with faint faces. When it closes, it goes straight into its lattice: the cubes around it as ghosts, tiling space (the lattice button hides or shows them). The closed square has its own lattice button too, showing the square tiling the plane. The indicator beside Wizard shows how far you've built (1D+, 1D+/2D, 1D+/2D/3D, 1D+/2D/3D/4D). ⊘ beside Undo clears the whole build to start again (Undo brings it back), and the Signal | Construct toggle under Wizard switches between the two 1D worlds.

Then the tesseract: when the cube closes (Open in 3D takes it to 3D's Cube piece), the fourth direction W rises from the start corner, pointing inward: W is drawn as perspective, so the second cube sits smaller inside the first. Build its first edge one cell per tap, then one tap per edge: the other seven W edges, then the inner cube. When it closes it goes straight into its lattice; tap to turn it slowly through W, the inner and outer cubes swapping places (tap again to stop), and Open in 4D takes it to 4D's hypercubic (Z4) world.

1D+ Construct · Kagome (Wizard → 1D+ → Construct · Kagome): first a hexagon, built by hand one cell per tap, its first side up the screen, clockwise, one continuous path. It takes Kagome's three directions: X, then Y at the first corner, then XY (X and Y together, so still 2D). Closing it is the 2D moment: its lattice button shows the honeycomb, and Open in 2D takes it to 2D's Hexagon tile. Then Kagome's star forms round it, one tap per edge (no new direction): out to each point and back to the next corner, where two of Kagome's lines cross. When the star closes it goes straight into the Kagome lattice, and Open in 2D takes it to the Kagome tile. Then Z, into pyrochlore (Kagome in 3D), in the same rhythm: first the hexagonal parts, the truncated tetrahedron standing on your hexagon (four hexagons, four triangles; Z's first edge by hand, then the rest in two taps), the 3D moment; then the mini tetrahedra on its triangles, all four in one tap, three on the star's points and one on top. Together they make one big tetrahedron, 3D's Pyrochlore piece: it goes straight into the pyrochlore lattice, and Open in 3D takes it there. Then W, into hyper-pyrochlore (Kagome in 4D), the same rhythm once more: the body, the truncated 5-cell (five truncated tetrahedra, yours one of them, and five tetrahedra; W drawn as perspective, inward), W's first edge by hand, the rest in two taps; then the limbs, a small 5-cell on each tetrahedron, all five in one tap. Together one big 5-cell: tap to turn it through W, and Open in 4D takes it to the Hyper-pyrochlore world. The body is cyan, the limbs gold, all the way up. Each Construct family (Square, Kagome, RD) keeps its own progress.

1D+ Construct · RD (Wizard → 1D+ → Construct · RD): the flagship. First the RD's own rhombus, by hand like the square, starting at its 70.53° corner, first side up the screen, clockwise: the 2D moment (its lattice button shows the rhombic lattice, and Open in 2D takes it to the Parallelogram at the RD Rhombus angle). Then Z into the rhombic dodecahedron, in the RD's own grammar: the body first, the cube inside it (cyan; its edges are the rhombi's short diagonals; Z's first edge by hand, then the rest in one tap), then the limbs, a pyramid on each of its six faces (gold, all six in one tap), whose edges are the RD's. It closes straight into its lattice, the twelve RDs round it, and Open in 3D places an RD there. Then W: the RD is the 24-cell's shadow, so splitting it open makes the 24-cell. Each corner of its cube parts in two, one on each side in W, and the two copies make the tesseract (the body: W's first edge by hand, the other corners at once). Then the limbs: the six apexes join both sides in one tap, and two new apexes out along W in another. That's the 24-cell, 96 edges, turning through W, with the RD left as its ghost; Open in 4D takes it to the D4 world.

2D: Parallelogram, Triangle, Hexagon, Kite and Kagome tiles, chosen from the 2D panel (at the top; at the bottom on a phone), each at up to four lattice angles (90°, 70.53°, 63.43° and 60°).

2D Kaleidoscope (Wizard → 2D → Kaleidoscope): a world of its own. Build with thick and thin Penrose rhombi, pentagons, triangles, hexagons and squares (all the same edge length, so any two fit edge to edge). Where you tap decides: near a piece's edge, the chosen piece goes across it; in empty space, it drops loose and slides along until it lines up beside a partner. Pieces never overlap. Mirrors reflect your build into a kaleidoscope: the mirror button switches between a Ring (○, 1 to 12 mirrors) and a triangle of three (△60°, △45°, △30°) repeating across the screen, set with the slider; Turn rotates your build against the mirrors, ↻ keeps it turning, ≈ shakes every piece loose to settle into a new pattern. Safe (Penrose shapes only) keeps the rhombi to Penrose's matching rule, so they always form a true Penrose tiling. Lattice View shows the mirror lines and, outlined, every spot the piece fits.

2D Nets (Wizard → 2D → Nets): from 2D to 3D. Pick a solid in the panel: the Voronoi cells, Cube, RD and TO (truncated octahedron), the space-filling cells of the simple, face-centred and body-centred cubic lattices; its net shows as a faint ghost. Follow it: tap to build the first face side by side, in 1D cells, then each tap builds the next face. When the net is complete, tap and it folds up into the solid (the indicator reads 2D/3D); the fold slider folds and unfolds it by hand, and Open in 3D takes the finished solid there. Long-press takes back a face (or unfolds). Each solid keeps its own progress. The Platonic solids are a second group: tetrahedron, octahedron, icosahedron, dodecahedron and the cube. Open in 3D appears for the solids the 3D world has as pieces (cube, RD, TO, octahedron). ⊘ beside Undo clears the net you're on.

Paint (every dimension but 1D+): the brush on the bottom row, or the Paint switch in the middle of the colour wheel, recolours pieces you've already placed. Turn it on, pick a colour, tap a piece; turn it off to build again. It switches colours to Pick, so each piece shows its own colour. (Shells and Golden Rhombohedra colour their pieces by band and type, so Paint isn't offered there, nor in Nets.) When the bottom row's slot is taken by an attach toggle (3D's Rhombohedra and Pyrochlore), the brush sits just above it; in 4D it's in the 4D panel.

3D:

4D:

Try this on FCC: place six Pyramids to form a Cube. Then add one Pyramid to each face of the Cube, and it becomes an RD. Remove those six again to go back to a Cube.

RD Quarter is one of the 4 rhombohedra an RD splits into. Tap an RD near one of its corners to fill that corner in, then tap a quarter's face to place its mirror image across that face. The mirror image always lands back on the RD lattice, so you can extend quarters from cell to cell. For a free rhombohedron lattice with Copy as well as Mirror, use Rhombohedra.

Colours

Pieces are coloured by the Colours setting (in Settings): Cyan (every piece cyan, the default), Type (each kind of piece in its own colour, editable in the list shown) or Pick (each piece keeps the colour it was placed with). Tap the Colour button (bottom left) to choose from 15 colours: in Cyan this switches to Pick, and in Type it changes the colour of the kind of piece you're placing. Switching never loses the colours pieces were placed with.

Looking at your build

Settings also has a Section view: pick an axis, drag the slider to move the cut, and tick Flip to see the other side.

Going into 4D

1. Choose 4D in the dimension picker, the Wizard, or Menu → Change Dimension.
2. Pick a world: Tesseract (Z4), 24-cell or 16-cell (D4), or a Hyper-pyrochlore piece.
3. Build as in 3D: tap a face to add the neighbouring 4D cell.

A 4D panel appears at the bottom of the screen.

• Slice / Projection switches how you see 4D. Slice (the default) shows the 3D cross-section at the current depth. Projection shows whole 4D cells as shadows; tap a shadow's face to build across it. In Projection you can also switch between Parallel and Perspective.
• The slider does what the button above it says: W-depth moves the slice through the fourth dimension, and XW, YW and ZW turn it into the fourth dimension. It clicks into place at useful stops. The one labelled FCC is the ordinary 3D RD world, and the one labelled Pyrochlore is the 3D Pyrochlore world.
• Reset 4D puts everything back to the starting position.
• Paint (the brush) appears in this panel for the 24-cell, 16-cell and Hyper-pyrochlore pieces, whose attach toggle takes the bottom row's paint slot.
• Info opens a panel showing the world and piece you're placing, what you've built, where the slice is (W-depth, or Projection), the XW/YW/ZW turn angles, and the 4D centre coordinates of the last cell you tapped or placed.

Going into 5D and 6D

5D and 6D are quasicrystals: a cube lattice in five or six dimensions, sliced through 3D space. The pieces fill space with no gaps, but the pattern never repeats.

• 6D is the icosahedral quasicrystal. Its pieces are two golden rhombohedra: prolate (tall) and oblate (flat).
• 5D is the decagonal quasicrystal: layers of the Penrose tiling, made of thick and thin rhombus prisms.

1. Choose 5D or 6D in the dimension picker, the Wizard, or Menu → Change Dimension.
2. Tap the cyan outline to place the first piece, then tap a face to add the neighbouring piece. In 5D, the top and bottom faces add a layer above or below.

You don't choose the shape: the tiling decides which piece goes in each slot.

A panel appears at the bottom of the screen.

• The slider does what the button above it says. Phason 1, Phason 2 and Phason 3 (6D only) move the slice sideways through the hidden dimensions: pieces flip, and some leave the slice while others come in. Pieces hidden by the slice aren't deleted; slide back and they return. Approximant clicks through periodic crystals (1/1, 2/1, 3/2, 5/3, 8/5 …) that get closer and closer to the true quasicrystal, τ, at the right-hand end.
• Build / Window switches to Window View, which shows the hidden dimensions. The window is a rhombic triacontahedron in 6D and a set of pentagons in 5D. Each corner of your pieces is a point: white inside the window (the corner is in the slice), red outside. Move a phason slider and watch points cross the window's edge as pieces appear and disappear.
• Reset 5D / Reset 6D puts the slider back to the start: phason 0 at τ.
• Info shows the world, what you've built, how many pieces are in the slice and how many it hides, the phason and approximant settings, and in Window View how many corners are inside. Items you've summoned are listed too: tap one to slide back to its settings.

Lattice View shows the surrounding tiling one step out as ghosts, and they flip as you slide a phason. World View works as in 3D.

The Catalogue

In 5D and 6D the bottom-left button opens the Catalogue (the Wizard's 5D and 6D cards open it too). It has four kinds of item:

• Zonohedra: shapes made of the tiling's own pieces, from a single rhombohedron up to the rhombic triacontahedron, and in 5D rhombus, hexagon, octagon and decagon prisms.
• Polytopes: the shadows of higher-dimensional polytopes, such as the 6-orthoplex, whose shadow is an icosahedron. A shadow lies over the tiling without blocking pieces, and its corners light up where they are in the slice. Long-press one to remove it.
• Bridges: hyperprisms, polytopes carried one step through a further dimension, such as the 5-orthoplex prism. Like polytopes, they land as shadows.
• Vertex stars: every way the pieces meet at a corner (7 in 5D, 24 in 6D), also with one or two rings of the pieces around them. Tap a heading to open its list.

Every item has a serial number. Type one in the box and tap Summon to go straight to it.

When you pick an item, a gold outline appears where it really occurs in your quasicrystal. Tap your build to move the outline to the nearest spot there, and tap the outline to place it as ordinary pieces. If the item belongs to a different approximant, the slider slides there first. Cancel summon stops. One Undo takes it back and slides the slider back to where it was.

Connect (in the panel, once you have two pieces) joins two items: tap a piece of each, and the shortest chain of real tiles between them is placed as one connector. One Undo takes it back.

Shells

A 3D world for building hulls: shells of rhombic dodecahedra (RDs), each shell its own colour band, counted out from your first piece. Choose it in the Wizard's 3D screen.

• + Shell fills the next shell, − Shell removes the outermost. Hull chooses how shells count: Steps (a cuboctahedron: 13, 55, 147 … pieces), Distance (toward a sphere), or a target shape (tetrahedron, cube, octahedron, RD or truncated octahedron). Trim cuts the hull exactly flat into that shape.
• Fragment mode: tap a piece, then pick a Breakdown (halves, thirds, quarters, sixths, eighths, twelfths, sixteenths, 24ths or 48ths); Turn changes the cut. In Build mode the Piece menu places single parts too.
• Scale builds with bigger RDs (×2 to ×4). Merge ×2 / ×3 on a targeted piece shows the big RD's outline, then Confirm swaps in one big RD; the Breakdown menu opens it again.
• Info shows the ring diagram: tap a ring to hide or show that shell and look inside, remove any shell, and pick the band colours.

Golden Rhombohedra

A 3D world for the two pieces of the 3D Penrose tiling. Choose it in the Wizard's 3D screen. Tap the cyan outline, then tap a face to add the Prolate or Oblate piece chosen in the Piece menu. Penrose check colours pieces green where they belong to the true aperiodic tiling and red where your build has drifted; Lattice View shows the true tiling around you, and tapping a ghost places it.

RHOMBIS

RHOMBIS is a 3D puzzle made from the same pieces: fill the target shape with the pieces in the tray. Open it from the welcome screen (Try RHOMBIS). Tap a piece in the tray, then tap a glowing void to place it; some pieces flip when tapped again. Stages lists every puzzle in sections and ticks the ones you've solved. ← Rhombiverse brings you back here.

Saving your work

Your World saves automatically in this browser, every dimension, after each change. It comes back when you reopen the site on the same device and browser.

In Settings:

• Export World saves everything, every 3D lattice, your 2D tiles and your 4D, 5D and 6D builds, to one file. Use it to keep a backup or move your World to another device.
• Import World opens an exported file. Undo takes an import back.
• Clear World (⊘ on the corner wheel) starts again with an empty world. Undo can bring it back.

Learning the maths

• Almanac: the maths and geometry behind every piece and lattice. Open it from Menu → Almanac.
• What's New lists recent changes.

Control reference

Screen buttons

Corner wheel

Drag the small wheel in the corner to turn it. Tap a face to use it.

Menu wheel

The menu is a rhombic dodecahedron. Each face is a section: tap a face to open it, and use Home to go back. Almanac is always on a top face.

Settings

Keyboard and mouse

Touch