DICTO User Guide

DICTO is the one menu over three apps by DICTO, built on one shared geometry. Rhombiverse is the landscape: the lattices themselves, stretching out in every direction, from 1D up to 6D. Polyhedraverse is the portrait gallery: the shapes that live in those lattices, one at a time, up close. Kaleidohedra moves the landscape: you can shear and slide the whole lattice, and every piece moves with it. Kaleidohedra and Rhombiverse are one app with two front doors, kaleidohedra.dictospheres.com and rhombiverse.dictospheres.com; Polyhedraverse has its own site for now.

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.

This guide starts with what Kaleidohedra and Rhombiverse share, then has a chapter for each app: Kaleidohedra, Rhombiverse and Polyhedraverse. The last part lists every control.

Getting started

Pick a dimension

After ENTER, DICTO opens on the dimensions, 1D+ to 6D, each showing the apps it holds. 3D+ is open to start with; tap another dimension to open it in place instead, or the open one to close it. 5D and 6D open their catalogues. 1D+ and 2D+ belong to no app: they are DICTO's own, in silver, the easier way in, and going up from them to 3D+ takes you into the app you came in by. An open dimension's content is listed in blocks, one per app in that app's colours, the app you came in by first, and nothing is listed twice. Tap a lattice, world or shape, each shown as a turning wireframe, to enter that app's space. The three app buttons at the top show one app only; tap it again to show all. DICTO (top left), Tab or Space bring it back at any time.

Place your first piece

An empty world shows an outline where the first piece goes, in the colour of the app you're in: cyan in Rhombiverse, orange in Kaleidohedra. 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

2D Nets (DICTO → 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. The Euclid–Kepler–Pacioli cell group has every piece of that cell, listed inside out: Pacioli's three golden rectangles, the icosahedron, octahedron, the stella octangula's two tetrahedra, cube, one spike of the great star, dodecahedron. Each folds into its true place beside the ones already folded, and Next goes on to the next piece. Whole cell shows everything you've folded together; tap to wrap it shell by shell, inside out, or drag the slider.

Paint: 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. Worlds that colour their pieces their own way (Shells, Golden Rhombohedra, EKP, Targets) don't offer it, nor do Nets and 1D+. When the bottom row's slot is taken by an attach toggle, the brush sits just above it; in 4D it's in the 4D panel.

3D+:

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.

Worlds in both apps

These 3D+ worlds are listed once in DICTO's 3D+, in Kaleidohedra's block, and open in Kaleidohedra's space.

Euclid–Kepler–Pacioli Cell Network (EKP)

A 3D world on one exact cell. Put Euclid's roofs on a cube and you get a regular dodecahedron; fold the roofs back in through the cube's faces and they make a regular icosahedron. The cube, dodecahedron and icosahedron edges are in the ratio φ² : φ : 1. Choose it in DICTO's 3D+ screen. Tap the outline, then pick a Piece (cube, dodecahedron, icosahedron, great stellated dodecahedron, octahedron, stella octangula or golden rectangles) and tap a solid: the piece goes into that cell if it isn't there yet, otherwise into the next cell across the tapped face. Smaller solids sit inside larger ones, so use X-ray or the translucent view to see them. View redraws the same build: two solids alternating by a Pattern, a checkerboard, or the merged outer surface of all the dodecahedra. Vertices marks the cube corners or every vertex, and Info shows the space group (Pm-3 for the cell itself; each pattern has its own). Adding dodecahedra across their faces only ever reaches cells of the same colour in the diagonal pattern; add across a cube face to reach the others. Turn odd cubes turns every piece in each odd cube a quarter turn about the vertical axis, so neighbouring cubes alternate. It also turns the merged outer surface.

Three of those pieces complete a homage to Kepler: the octahedron on the cube's face centres, with the icosahedron's corners on its edges at the golden section; the stella octangula, two regular tetrahedra on alternate cube corners that overlap in that octahedron; and Pacioli's three interlocking golden rectangles, which are exactly where the neighbouring cells' roof ridges meet. Together all five Platonic solids nest in one cell: icosahedron, octahedron, tetrahedra, cube, dodecahedron.

The Dogstar is a piece too, between the octahedron and the stella octangula in the wrap order: the hole regular dodecahedra leave in their densest packing (George W. Hart's stellation 8 of the dodecahedron, 1996), inside the stella and holding the next cell's dodecahedron, 1/φ³ the size. In 2D+ → Nets it unfolds into one 60-triangle net. Dodecahedron / Dogstar fills the gaps the usual way: dodecahedra on the even cells and Dogstars in the gaps between them, the Sunstar Lattice's tiling. Tetrahedral clusters, Octahedral clusters and Octet network work as in the Stella–Jewel Lattice, with dodecahedra: six round an odd cell hold its Dogstar inside, a Sunstar turned inside out. Kagome network: up and down tetrahedral clusters sharing single Jewels at their corners, Kagome style (gold up, violet down), with a single Jewel in each cell between them, where a violet flame burns. In every cluster arrangement each cluster shows as one solid block, the pieces between the clusters fill in by themselves (stellas, and in the Kagome network single Jewels in violet), and each tap places the next cluster where you tap. Both worlds also have the six Kagome hull networks (octahedral, rhombohedral, cuboctahedral hollow and solid, cubic, rhombic dodecahedral), clusters sharing single pieces corner to corner; and the Stella–Jewel Lattice has the DICTO Hexa diamond network and the DICTO Hexa and Hexa-Key checkerboard, where Hexas meet window to window and a Hexa-Key fills each gap once four Hexas surround it.

The cell behind this world is the Euclid–Kepler–Pacioli cell, and great stellated dodecahedra and icosahedra touching only at corners form the Euclid–Kepler–Pacioli network, both by DICTO. Cite them as doi:10.5281/zenodo.23173809.

Stella–Jewel Lattice

A 3D+ world (DICTO → 3D+ → Stella–Jewel Lattice) of two pieces that fill space together in a checkerboard: the DICTO Jewel (DJ, DICTO's name for the windows solid of the Studies world) on the even cells and the stella octangula on the odd cells. Tap the glowing DICTO Jewel to place it, then tap any face to add the piece across it; long-press to remove. View switches between both pieces and DICTO Jewels alone, which meet face to face on all 12 of their rhombi and leave stella-shaped holes. Shear works as in Studies: exact copies on the moving cell centres, or the whole packing bent. Five-fold axes draws each DICTO Jewel's six five-fold axes and, on each face, the five places a window could sit, the one the cube picks bright. Lattice View shows the empty cells you can fill next. Jewel chain shows inside each DICTO Jewel the chain that nests in it, every step touching: its cube, its stella octangula, a Dogstar, the Dogstar's core dodecahedron (1/φ³) and the next DICTO Jewel, 1/φ³ the size. Tetrahedral clusters and Octahedral clusters (each its own entry in DICTO's 3D+, like the octet and Kagome networks) place DICTO's clusters whole, a tap each: four DICTO Jewels in a tetrahedron, touching at one point in the middle, or six round a stella octangula, solid. Both fill space with stella octangulas. Octet network shows the clusters sharing Jewels as the cells of the octet truss: build with Jewels, see-through, and each complete tetrahedral cluster shows as an amber tetrahedron, each octahedral one as a blue octahedron, its stella inside.

Sunstar Lattice

A 3D+ world (DICTO → 3D+ → Sunstar Lattice) of regular dodecahedra in their densest lattice packing on the even cells and, in each hole they leave on the odd cells, a Dogstar: an 8-pointed star, exactly a partial stellation of a dodecahedron 1/φ³ their size, with only golden edge lengths. A dodecahedron with the Dogstars round it is a Sunstar, the sun with its sun dogs (DICTO's names). Tap the glowing dodecahedron to place it, then tap any face to add the piece across it; long-press to remove. View shows both, the Dogstars alone (they share corners, four at each cube corner, like a 3D Kagome; a tap adds the nearest Dogstar sharing a point) or the dodecahedra alone. Shear and Five-fold axes work as in the Stella–Jewel Lattice. Whole Sunstars adds and removes a dodecahedron together with the 6 Dogstars on its faces (a Dogstar another Sunstar still has stays). Dogstars in every cell puts a Dogstar inside each dodecahedron too (it fits wholly inside: Dogstar inside stella inside cube inside dodecahedron), the dodecahedra see-through, so Dogstars fill every cell with eight tips meeting at each cube corner. Star Chain Reaction (DICTO's name, like the Sun's own chain reaction) shows inside each dodecahedron the chain that nests in it exactly, each step touching: its great star (the great stellated dodecahedron of the Dogstar's core), inside that a whole Sunstar 1/φ³ the size, inside its dodecahedron the next great star, and a Sunstar 1/φ⁶ the size. The Dogstar is George W. Hart's stellation 8 of the dodecahedron (1996), which he noted fills space alternated with regular dodecahedra; Polyhedra-World also shows it. Hans Walser describes it too, as a cube with its six hip roofs cut away (his “semi-regular dodecahedron”), which fills space with regular dodecahedra like a 3D chessboard.

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 DICTO'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 DICTO's 3D+ screen. Tap the 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.

The same two blocks are the pieces of DICTO's RHOMBITURE, an extractable armature system for carving and modelling: doi:10.5281/zenodo.23173896.

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 (⊘, top right) starts again with an empty world. Undo can bring it back.

Learning the maths

• Almanac (📖, top right): the maths and geometry behind every piece and lattice.
• What's New (🕘) lists recent changes, and About (ℹ) shows the welcome screen again.

Kaleidohedra

In Kaleidohedra you build on a lattice and then move the lattice itself: the ⟋ Shear panel slides and bends it, and every piece you placed moves with it. It has the 3D+ lattices, DICTO's worlds, Nets (2D+), and two worlds of its own, Studies and Targets. 1D+, 4D, 5D and 6D are Rhombiverse's.

Shearing the lattice

Kaleidohedra's own controls are in the ⟋ Shear panel at the top right. They move the whole lattice at once: everything you've built slides with it, and building carries on as usual.

• ↺ Reset to FCC takes everything back to the ordinary FCC lattice with its regular rhombic dodecahedron, if you lose your way.
• Towards picks where the lattice is heading: DICTO FCC (from the ordinary FCC lattice to DICTO's sheared one) or Bain (BCC → FCC).
• Path slides along that exact route; the stop buttons (such as FCC, halfway and DICTO FCC) jump to its named points.
• Cell changes the cell's shape on the same lattice: 0 is the plain sheared cell, 1 the cell with all edges equal (at the DICTO FCC stop, DICTO's skewed rhombic dodecahedron). A red band under the Path slider marks where, at this Cell value, the cell no longer fills space (near the path's ends); there the readout says so too. Cell 0 always fills space.
• The Regularity meter scores the cell by its angles (1 means every angle is special: 36, 45, 60, 70.5, 72 or 90°). On the Bain path it also counts how many of its disphenoids are regular tetrahedra.
• Find jumps along the path to the next most regular cell or the next moment a regular hexagon appears, round to the first after the last. Tap a stop first.
• Six sliders set the lattice's lengths a, b, c and angles α, β, γ directly.
• Export member saves the current state under a name you choose, as a small JSON file (a population member).
• ☆ Keep keeps the current state in your own Kept list, in this browser only. It stores the state's ID and a fingerprint, never the shape itself: Open regenerates it from the ID, and a mark shows whether it came back the same (✓), the app now makes a different shape for it (⚠), or it can't be made any more (✗). Copy ID copies the ID, to send to DICTO if you think it belongs in the record.

In Euclid–Kepler–Pacioli Cell Network the Shear panel moves the lattice only: the cell centres slide, and every piece stays a true regular solid. It hides in Targets, where shearing would change the angles. The findings behind Kaleidohedra, and how each one is checked, are in DISCOVERIES.md.

Studies

A 3D+ world of exact constructions on the Euclid–Kepler–Pacioli cell, shown one at a time (DICTO → 3D+ → Studies). Pick a Study: Windows (the dodecahedron with its six face-neighbours' stella octangulas carved out, leaving 12 Penrose thick rhombi at the dodecahedron's own face angles), Windows, with the six stellas, Dodecahedra and their Dogstars (regular dodecahedra on the even cells leave one hole in each odd cell: the Dogstar, an 8-pointed star, a partial stellation of a dodecahedron 1/φ³ their size, inside the cell's stella octangula; together they fill space, and Apart pulls them apart), Windows and stellas, checkerboard (the stella octangula is the windows' other half: windows in the even cells and stellas in the odd ones fill space exactly, 12 + 4 = two cubes; the Apart slider pulls them apart to see them fit), Windows made convex (each rhombus pushed straight out; the Push slider snaps at the golden point, where 6 golden rhombi appear, 74 faces in all), Windows into the icosidodecahedron and Windows into the rhombic dodecahedron (the Morph slider moves the rhombi into each; the hull is drawn solid with the rhombi inlaid, their outlines showing through it on the way), and Stretched dodecahedron (8 pentagons, 4 hexagons and 2 rectangles; the Stretch slider snaps at one edge, where the rectangles are squares, and at the lattice spacing). Studies shear too: as exact copies on the lattice or, one tap away, as the solid itself.

Targets

A 3D gallery of the 160 target cells: every space-filling cell with all edges equal whose edges meet only at special angles (36°, 45°, 60°, 70.53°, 72° and 90°), found by an exact search done two ways (see TARGETS.md). Choose it in DICTO's 3D+ screen. Pick a Type (parallelepiped, hexagonal prism, rhombic dodecahedron, elongated dodecahedron or truncated octahedron) and a cell. Each is named by its type, its number in TARGETS.md and its angles; ✓ marks the ones already in Polyhedraverse. Show draws the cell alone, with its face neighbours, or as a 3×3×3 block of its lattice, so you can see it fill space. Faces are coloured by kind: squares blue, rhombi pink, regular hexagons gold, other hexagons purple. Info lists its faces, the angles between its edge directions, its volume (edge 1), the lattice it tiles as, and whether it is in Polyhedraverse yet. Nothing is built here, and the Shear panel hides, so the angles stay exact.

Rhombiverse

In Rhombiverse you build with shapes that fill space, from a line (1D+) to quasicrystals (5D, 6D), tapping a face to add the next piece. Its own worlds are 1D+, 2D+, 4D, 5D, 6D and the RHOMBIS puzzle.

1D+

1D+ Signal (DICTO → 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. 🔇 / 🔊 turns on the sound: a soft hiss of deep-space static, with the dots and dashes as tones, in time with the letters arriving (off to start with; remembered on this device; only in Signal).

1D+ Construct (DICTO → 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 DICTO 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 DICTO 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 (DICTO → 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 (DICTO → 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+

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 (DICTO → 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.

Going into 4D

1. Choose 4D in DICTO.
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, W: flat and W: deep switch how the fourth dimension is drawn (flat, or with depth); the 3D / ∥ / ISO button in the tools column sets the camera, as everywhere.
• 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.

4D:

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 DICTO.
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 (DICTO'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.

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.

Polyhedraverse

Getting started

Polyhedraverse is the portrait gallery: every shape in its families, one at a time, up close. Turn it, compare it, attach shapes face to face or vertex to vertex, and print any shape's net to fold. Its space is inside DICTO: choose 3D+, then a family in Polyhedraverse's green block, then a shape, and tap the green outline to place it. Tap attach: tap a face, then a shape in the strip that appears (More… lists every shape that fits); it goes on at its best fit and stays chosen, so each face you tap next gets it in one tap (✕ to stop). Long-press removes a piece. It opens from DICTO in any of the apps.

The shape browser

At the top of Polyhedraverse's block in DICTO, search finds shapes by name, family, face shape (e.g. pentagon) or number of faces, and Favourites and Recent list the shapes you've starred or used. Tap a shape for its details: a turning preview, its families, its faces by kind, edges, vertices, whether it is convex and what it pairs with, with Build with it and ☆ Favourite (the same as pinning it in the strip). Shapes with a printable net show Net in their details: fold it up with the slider or Fold up, and Download PDF (A4), glue pairs numbered alike, with or without tabs. Compare puts two shapes side by side, with the faces they share (where one can go on the other). 4D prism shows a shape's 4D prism as a 3D shadow, and for the shapes that are cells of a 4D polytope, Extend into 4D shows the whole polytope, every cell in perspective. The Kepler–Poinsot group has the four star solids with their real star faces, Schläfli symbol and density.

The families

Attaching

When a build has two or more pieces, its name shows under 3D+ (named builds such as DICTO-Star or Stella Octangula are recognised); tap it to read all of it. The strip keeps the last 8 shapes you used, saved with the build; long-press a shape to pin it so it is always offered (long-press again to unpin).

Tap near a corner instead of a face and a dot marks the corner: any shape then goes on there, corner to corner, pointing straight out, turned by itself to the best fit. With a face tapped, 4D Prism (on shapes that have one) adds the same shape pushed straight out, joined by a see-through prism cell, and Transform to… turns a 10- or 12-face deltahedron into the other in place. A build of golden rhombohedra shows the Golden helper: how many pieces sit in the true 3D Penrose tiling, Next safe piece, and the golden zonohedra built step by step.

Hints: what comes next

Hints: what comes next. Build with a piece that has a known pattern and faint ghosts show where the next pieces go, around the piece you placed last, each kind in its own colour; the faces they would touch glow in the same colour, so matching faces share a colour. The patterns: the DICTO Hexa and Hexa-Key checkerboard; the DICTO Dodeca-13's mirror column and bcc stack; and every space-filling pair's honeycomb (the Sunstar and Stella–Jewel Lattices, the octet truss, pyrochlore, rectified and truncated cubic, and the three prism honeycombs). A piece that belongs to several pairs (tetrahedron, octahedron, cube, triangular prism) shows them once its partner is in the build. Tap a ghost or a glowing face to place that piece there. Hints in the strip turns them off, for attaching anything else. A piece never goes where it would pass through one already built; if nothing fits, it says so.

DICTO's pieces

DICTO's pieces start with the DICTO Hexa family: the DICTO Hexa (eight DICTO Jewels in a cube, passing through each other inside, one solid), the DICTO Hexa-Key (the gap between Hexas: Hexas and Hexa-Keys fill space together), two clusters of eight Hexas (rhombohedral, with a Hexa-Key sealed inside, and diamond, sharing corner Jewels), and the trimmed Jewel and roof they split into. With one in the build, Parts shows it as its Jewels, or split into pieces that don't overlap: A (cubes and roofs) or B (whole and trimmed Jewels).

DICTO Dodeca-13. In DICTO's pieces: DICTO's thirteen dodecahedra (one in the centre, one on each face) with the gaps filled, as one solid, plus its Star, Unit, Wedge and Needle as pieces of their own. All attach by their faces. With a Dodeca-13 in the build, Parts shows it as pieces (13 dodecahedra, 30 cyan wedges, 20 purple needles), as units (the centre and 12 units that lift straight out) or as stars (overlapping, see-through).

Building in 4D

4D build. Place a tetrahedron, cube, octahedron or dodecahedron on its own and tap Grow into 4D: it grows into a 4D polytope's 3D shadow (the 5-, 16- or 600-cell, the tesseract, the 24-cell or the 120-cell). Shell 1 comes a cell at a time, then a whole shell per tap; Open shows shell 1 as true copies on the faces, Closed where they really sit (from shell 2 on it stays closed); RCP-Coordinates marks each cell's 4D coordinate. DICTO → 4D → 4D Polytopes shows the six regular polytopes; Build starts one from its seed.

Saving your work

Your build saves itself as you go, separately for each app's address, and Undo (hold to scrub) takes back any step. Settings → Export / Import World moves builds between devices and addresses. On polyhedraverse.dictospheres.com, a build saved by the earlier Polyhedraverse site is brought over the first time you visit.

Control reference

Screen buttons

Tools (top right)

A column of buttons in the top right corner, in the colour of the app whose space you're in.

DICTO wizard

DICTO (top left) opens the wizard over the whole screen, in the colours of the app you came in by. 3D+ is open to start with; tap another dimension to open it in place instead, or the open one to close it. 5D and 6D open their catalogues. 1D+ and 2D+ belong to no app: they are DICTO's own, in silver, the easier way in, and going up from them to 3D+ takes you into the app you came in by. It starts on the dimensions, 1D+ to 6D, each showing the apps it holds. In a dimension, each app has its own block in its own colours, the app you came in by first, and anything already listed is not repeated. Choosing anything takes you into that app's space: its colours, and its tools (Kaleidohedra's Shear appears only there). The three app buttons at the top are a filter: tap one to show only that app, tap it again to show all. In 3D+, DICTO's own lattices and worlds come first, in a silver DICTO block (EKP, Studies, Stella–Jewel, Sunstar, Targets, Shells, Golden, DICTO FCC, the DICTO Hex Prism), and open in the app you came in by (the DICTO Hex Prism in Rhombiverse); the other app, whose lattices are the same, keeps one row: Kaleidohedra's Shear any lattice, or Rhombiverse's Plain lattices.

Settings

Keyboard and mouse

Touch