01 / The fourth spatial dimension
Tesseract Explorer
Rotate a tesseract in 4D and move a 3D slice along W to watch the cross-section change shape.
Open the explorer →0D
A point
Every journey starts somewhere, and this one starts with a point. It has no length, no width and no direction, only a position. There's only one kind of point, so 0D has exactly one regular shape.
1D
A line segment
Pull the point in one direction and it traces out a line segment, with a point at each end. Every segment is the same shape, only longer or shorter, so 1D also has exactly one.
2D
The regular polygons
Now join segments end to end, turning a corner at each join, until the path closes into a loop. That's a polygon.
A regular polygon has sides of equal length and equal corner angles. Here that always means a convex polygon, which never doubles back on itself: a five-pointed star also has equal sides and angles, but its edges cross, so it isn't counted here (stars get a bonus section at the end).
You can make a polygon with any number of sides, so 2D has infinitely many. As the number of sides p grows, each interior angle, (p − 2) × 180° ÷ p, creeps toward 180° and the shape approaches a circle. You can think of a circle as the end of that road: a polygon with infinitely many sides, each infinitely short. Strictly, a circle never quite gets there, since it has no straight sides or corners at all, which is why it isn't counted among the polygons.
The interior angle (the highlighted arc) is what decides which polygons can build a solid.
2D → 3D
Folding polygons into solids
To leave the flat plane, polygons have to tilt up out of it. To make a corner of a solid, gather identical polygons around one point. Laid flat, they leave a gap (highlighted). Fold them up until the gap closes and their edges meet. Repeat that same corner everywhere, and the surface closes up into a Platonic solid, shown faintly once the corner snaps shut.
If the polygons already fill 360° when flat, there's no gap to close and they can't fold. They tile the plane instead. More polygons than that won't even fit around the point.
Drag any figure to turn it by hand.
Only three polygons make it past 2D
Triangles , squares and pentagons are the only regular polygons whose corners leave a gap to close, so they are the only ones that can fold up into a solid. Hexagons fill 360° exactly and lie flat, and polygons with more sides overflow.
Every regular polytope in 3D and higher is built from these three: look at any 2D face of any shape further down this page and it will be a triangle , square or pentagon . The small symbol in the top-left corner of each shape's card shows which. Pentagons drop out after 4D, so from 5D on only triangles and squares remain.
3D
The five Platonic solids
Repeat each of those five corners all the way around and the surface closes up into a solid. Each solid is built from one kind of regular polygon, with the same number meeting at every corner. Plato tied four of them to the classical elements and the dodecahedron to the heavens. Euclid closed his Elements by proving that no other convex solids exist.
Drag any figure to turn it by hand.
Why only five?
At each corner the face angles must add up to less than 360°. If they add up to exactly 360°, the faces lie flat and you get a tiling. If they add up to more, the faces can't fit around a point. Only five combinations stay under the limit.
| Faces at a corner | Angle sum | Result |
|---|---|---|
| 3 triangles | 180° | Tetrahedron |
| 4 triangles | 240° | Octahedron |
| 5 triangles | 300° | Icosahedron |
| 6 triangles | 360° | Flat tiling |
| 3 squares | 270° | Cube |
| 4 squares | 360° | Flat tiling |
| 3 pentagons | 324° | Dodecahedron |
| 3 hexagons | 360° | Flat tiling |
Reading the labels
Schläfli symbol {p,q} means q regular p-gons meet at each vertex. So the cube is {4,3}: three squares at every corner, and each square has four vertices (a 4-gon).
Duals swap faces and vertices. Put a dot at the center of every face of a cube and connect them, and you get an octahedron. The dual of {p,q} is {q,p}. The tetrahedron {3,3} is its own dual.
Euler's formula
Count a solid's corners (V, for vertices), its edges (E) and its faces (F). Leonhard Euler noticed in 1750 that for every convex solid, V − E + F comes out to exactly 2. A cube has 8 − 12 + 6 = 2. A dodecahedron has 20 − 30 + 12 = 2. Each card above lists its V, E and F, so you can check the rest.
Size and shape don't matter, only how the surface joins up. Draw a new edge across a face and you add one edge and one face, which cancel. Put a new corner in the middle of an edge and you add one corner and one edge, which also cancel. However you reshape or subdivide the surface, the total stays 2. That 2 belongs to every surface shaped like a sphere.
A surface with holes gives a smaller number. A solid shaped like a picture frame, with one hole through it, gives 0, and each extra hole takes away another 2. Two of the star solids give −6, and in 4D the formula gains a term for cells.
3D → 4D
Three ways to climb a dimension
We can't fold solids into 4D by hand (though we can watch the shadow of a fold), so how do we get there? Look back at how we came. Three of the Platonic solids can be built up from a single point by repeating one simple move, adding a dimension each time. Make the same move once more and it carries each of them into 4D. It keeps working after that too, which is why these three families exist in every dimension.
2D → 3D
Drag any figure to turn it by hand. Hold Shift, or switch on Drag through W, then drag the 4D shapes at the end of each row left or right to turn them through the fourth dimension.
Looking straight down the W axis
In the build animations above, while each shape climbs into 4D, the new fourth direction, W, is drawn at a slight slant so you can watch points move along it. At the end, the view swings round to look straight down W. Seen that way, anything further along W simply looks smaller and sits in the middle. The tesseract becomes the familiar cube inside a cube, and the 5-cell becomes a tetrahedron with its fifth point at the centre, joined to every corner.
The 16-cell hides something in this view. Its two new points sit at opposite ends of the W axis, so looking straight down it stacks them exactly on top of each other at the centre. Each of them joins all six corners of the octahedron, and those twelve edges overlap in pairs, so you see six spokes lining up into three straight lines through the middle: the octahedron with its three axes drawn in. The drawing shows 6 points and 18 lines, but the 16-cell really has 8 vertices and 24 edges. A quarter of it is hidden behind the rest.
Regular isn't the only way up
These three moves work on any shape, not just regular ones. Slide a copy of a triangle and you get a triangular prism. Slide a dodecahedron into 4D and you get a dodecahedral prism. Add a point above a square and you get a square pyramid. The results are perfectly good shapes one dimension up, just not regular, because their faces and corners don't all match.
Starting from a single point and repeating the same move every time keeps each step regular, which is why these three families go on forever. Not every regular shape comes from these moves, though. The dodecahedron and icosahedron, the 120-cell and 600-cell, and the 24-cell are built differently, and none of these exceptional families extends to 5D or higher.
Relax the rules a little and higher dimensions fill up fast. 3D has 13 Archimedean solids, like the soccer ball, whose faces are regular polygons of more than one kind and whose corners are all alike. In 4D there are 47 such convex shapes, not counting prisms, and only six of them are regular. 4D also has infinitely many duoprisms, made by "multiplying" two polygons, such as a triangle by a pentagon. The regular shapes are the rarest, most symmetrical few.
4D
The six regular polychora
This is where the journey leaves the world we can see. In 4D the building blocks are regular solids, called cells, glued face to face around every edge. Ludwig Schläfli found all six around 1852. Five of them are direct analogues of the Platonic solids. The sixth, the 24-cell, has no counterpart in any other dimension.
These drawings are perspective shadows: the 4D shape is rotated in 4D, then projected down to 3D and onto the screen, the same way a cube's shadow can look like a square inside a square.
Drag any figure to turn it by hand. Hold Shift, or switch on Drag through W, then drag left or right to turn it through the fourth dimension.
The 3D ↔ 4D family tree
| 3D solid | 4D analogue | Its cells |
|---|---|---|
| Tetrahedron | 5-cell | Tetrahedra |
| Cube | Tesseract | Cubes |
| Octahedron | 16-cell | Tetrahedra |
| Dodecahedron | 120-cell | Dodecahedra |
| Icosahedron | 600-cell | Tetrahedra |
| None | 24-cell | Octahedra |
The counting rule in 4D
With C for cells, Euler's formula becomes V − E + F − C = 0. For the tesseract that's 16 − 32 + 24 − 8 = 0.
That 0 isn't a mistake: the answer alternates with dimension. A 2D polygon gives V − E = 0 (a square has 4 − 4), a 3D solid gives 2, a 4D shape gives 0, and a 5D shape gives 2 again.
The symbols gain a third number. {p,q,r} means r of the {p,q} cells meet around each edge. The tesseract is {4,3,3}: three cubes around every edge.
How rotation works in 4D
Rotation really happens in a plane, not around an axis. When a top spins, its X and Z directions are being turned into each other. 3D has three such planes: XY, XZ and YZ. 4D has six: those three, plus three that each mix the fourth direction, W, with X, Y or Z.
Think of a cube's shadow on a wall. Spin the cube flat against the wall and the shadow just spins. Tip it toward the wall and the near face shrinks while the far face grows, until they swap places. The cube never bends, but its shadow turns inside out. Turning a tesseract through W does the same one dimension up: the small inner cube, small only because it is further away in W, grows and moves outward while the outer cube shrinks inward. They pass through each other and swap, and the picture is a cube inside a cube again. In 4D every edge keeps its length and every corner stays square throughout. Only the shadow distorts.
Slice it in Hyperslice ↗Turning the 4D shapes
Every 4D shape on this page, in the cards above and at the end of each climb in Three ways to climb a dimension, moves in two separate ways, and each has its own button. Pause 3D rotation controls an ordinary spin, which only turns the shadow, like a solid object turning in your hand. Auto-turn through W turns the shape through the fourth dimension, which changes the shadow's shape: the parts further along W, drawn smaller and nearer the middle, swell outward as the outer parts shrink inward, until they have traded places. Pause the 3D rotation to watch that on its own. Drag through W, or holding Shift as you drag, does it by hand. The shape itself never bends or stretches; only its shadow distorts.
4D, unfolded
Folding nets into 4D
Cut a solid along some of its edges and it opens out flat into a net: a cube into a cross of six squares. Fold the squares back up and the cube closes. A 4D shape can be cut open the same way, into a net of solids laid out in ordinary 3D space. Fold each solid up about the face it shares with its neighbour and the 4D shape closes. They all fold into W, a direction none of us can point to, so we can only watch the shadow. Below, each of the six regular 4D shapes from the section above is folded up from its net.
Each row pairs a fold we can see, a solid folding up in 3D, with its 4D cousin, seen straight down W. In that view anything further along W looks smaller and sits nearer the middle, so a cell folding away seems to shrink and slide inward, though in 4D it stays a perfect, rigid solid. The cell to follow is drawn in cyan , and cracks that have yet to close are highlighted.
Drag any figure to turn it.
5D and up
Only three, forever
From 5D upward, the angle budget gets so tight that only three families survive, and they exist in every dimension n. These are the descendants of the tetrahedron, the cube and the octahedron. The dodecahedron's line ends at the 120-cell and the icosahedron's at the 600-cell.
Like the 4D drawings, these are perspective shadows: each hidden direction is projected away in turn, so parts further along it look smaller and sit inside the rest. Use the slider to choose the dimension. As in 4D, Auto-turn and Drag through turn a shape through just one hidden direction (the one picked in the menu), so what you see change comes only from that dimension.
5
Pick a hidden axis (or hold Shift for the 4th), then drag left or right to turn the shape through it.
The end of the road
From here on, nothing new ever appears. Climb to 10D or 1,000D and you'll find the same three: a simplex, a cube and an orthoplex, the distant descendants of the triangle , the square and the square (diamond) from 2D. The richest stops on the journey are 3D and 4D, where the five Platonic solids and their six cousins live.
Bonus
Beyond convex: the star polytopes
The journey above is complete, but it only followed convex shapes. Any shape that isn't convex is called non-convex, and there are two ways to be one:
- Concave : the outline has a dent. At least one corner points inward, like the notch at the back of an arrowhead, but the edges never cross.
- Self-intersecting, or star : the edges cross through each other, like the lines of a pentagram. In 3D the boundary is made of faces, so it's the faces that pass through each other. The great dodecahedron's edges never cross at all, but its pentagons slice through one another. In 4D, the cells do.
Under the classical definition of regular that this page uses (a single, finite shape with flat faces), a regular shape can't be concave. Every corner of a regular shape is the same, so if one corner pointed inward, they all would. A closed outline can't do that without crossing itself, and once it crosses itself it's a star. So every regular non-convex shape is a star, and stars are what this section is about. Looser definitions of regular allow more exotic objects, such as infinite or zigzagging "skew" polyhedra, but those are beyond this page.
This section counts only the regular ones. Non-convex shapes in general are endless, from any dented polygon to the 53 other uniform star polyhedra, which mix more than one kind of face. Allow regular stars, and the count at each stop changes:
2D: star polygons
Every star polygon is drawn from a recipe of two numbers:
- Place p points evenly around a circle.
- From any point, draw a line to the point q steps along. Keep going until you're back where you started.
The result is written {p/q}. With q = 1 you join neighbours and get the ordinary polygon: {5/1} is just the pentagon. With five points and q = 2, each line skips a point, the lines cross, and you get the pentagram {5/2} , the smallest star polygon. Every edge is the same length and every point has the same angle, so each one is regular.
Two rules keep a recipe valid:
- q must be less than half of p. Stepping q points one way is the same as stepping p − q the other, so {5/3} is just {5/2} drawn backwards.
- p and q must share no common factor. Otherwise the path gets back to its start before reaching every point. {6/2} visits every other point and closes after three lines as a triangle, leaving the other three points to form a second triangle. That's two shapes, not one star.
Six points have no star at all, but every other number of points from 5 up has at least one, and larger numbers have several: seven points give both {7/2} and {7/3}. So, just like convex polygons, there are infinitely many. The angle under each star below is the angle at each of its points.
3D: the four Kepler–Poinsot solids
A Platonic solid is set by two things: the shape of its faces, and how the faces gather around each corner. In a star solid, one of the two is a star:
- Star faces: every face is a pentagram . That gives the small stellated dodecahedron and the great stellated dodecahedron.
- Star corners: the faces are ordinary pentagons or triangles , but instead of going once around each corner, five of them go around it twice, crossing over each other like the lines of a pentagram. That gives the great dodecahedron and the great icosahedron.
Either way, the faces have to pass through each other to fit together. There are exactly four of these solids. The symbol in the corner of each card below shows its face shape.
None of them needs new corners. Three sit on the icosahedron's 12 corners and one on the dodecahedron's 20, joined up in a different way.
Johannes Kepler described the two with star faces around 1619, though one had already appeared in a 15th-century floor inlay in St Mark's Basilica in Venice, sometimes credited to Paolo Uccello. Louis Poinsot found the two with star corners in 1809, and three years later Augustin Cauchy proved there are no more.
Folding a star corner
All four corners are built on the pentagram, but in two different ways. Two of them use pentagram faces. The other two use ordinary pentagons or triangles, arranged around the corner in a pentagram.
To see that second kind, look straight down at one corner of an ordinary icosahedron. Its five triangles go once around the point in order, and joining their outer ends draws a pentagon. At a corner of the great icosahedron, each triangle skips over its neighbour, the same way {5/2} skips every other point of five. The five triangles go around the point twice before they close, and joining their outer ends in that order draws a pentagram. The great dodecahedron does the same with pentagons.
Each of these corners folds up from flat faces, just like the Platonic solids'. Gather the faces around a point and fold them up until the gap (highlighted) closes:
- Star faces: five pentagrams meet at each corner of the small stellated dodecahedron, and three at each corner of the great stellated dodecahedron. Each pentagram point is only 36°, so five of them cover 180° and three cover just 108°, leaving a wide gap. Folding it shut makes a sharp, spiky corner.
- Star corners: the faces have to go around the point twice, so the gap is measured against two full turns, 720°, instead of one. It's drawn as a spiral so you can see its second lap. Five pentagons add up to 540°, already more than one full turn, so they overlap when laid flat. Five triangles add up to only 300°. They fold through the ordinary icosahedron's corner on the way, then keep going until they close on the second lap.
Once a corner closes, the solid's other faces slide in, ring by ring: the same corner, repeated at every point. Then the finished star solid stays behind faintly.
Drag any figure to turn it by hand.
Drag any figure to turn it by hand. The faces are see-through, so you can follow them as they pass through each other.
Reading star symbols
A fraction in a Schläfli symbol means a star. The top number is how many points sit evenly around a circle, and the bottom is how many places you step along when joining them. {5/2} joins every second point of five, so the lines cross into a pentagram and wind twice round the middle.
The small stellated dodecahedron is {5/2,5}: pentagram faces , five at each corner.
The great dodecahedron is {5,5/2}: ordinary pentagons , five at each corner, but the five wind twice around it, like the points of a pentagram .
Duals still swap the numbers. {5/2,5} and {5,5/2} are a dual pair, and so are {5/2,3} and {3,5/2}.
When Euler's formula stops giving 2
For the small stellated dodecahedron and the great dodecahedron, V − E + F = 12 − 30 + 12 = −6, not 2. Euler's 2 only holds for surfaces shaped like a sphere (see how it works), and these two aren't.
Their faces pass through each other, but the lines where one face cuts through another aren't real edges. They're just places where the surface crosses itself. Imagine pulling the faces apart so they stop crossing, while keeping each face joined to the same neighbours along the same edges. A convex solid would come out shaped like a sphere. These two come out as a surface with four holes, like a pretzel with four loops. You can't see the holes in the model, because building that surface from flat, regular faces forces it to pass through itself and hides them.
Each hole takes 2 off Euler's total, so four holes give 2 − 4 × 2 = −6. Pulled apart the same way, the other two star solids come out shaped like a sphere, so they still give 2.
4D: the ten Schläfli–Hess polychora
A 4D shape's symbol {p,q,r} describes three things: its faces {p}, its cells {p,q}, and how many cells gather around each edge, r. In a star polychoron, a 5/2 appears in one of those slots, and where it sits tells you where the star is:
- First slot, star faces: every face is a pentagram , so the cells are star solids too. For example, the small stellated 120-cell {5/2,5,3}.
- Middle slot, star cells: the faces are ordinary, but the cells are star-cornered solids, great dodecahedra or great icosahedra. For example, the great 120-cell {5,5/2,5}.
- Last slot, star edges: the cells are ordinary Platonic solids, but five of them wind twice around each edge, like the lines of a pentagram. For example, the grand 120-cell {5,3,5/2}, made of plain dodecahedra.
One, the grand stellated 120-cell {5/2,5,5/2}, has stars in two slots. As in 3D, none of them needs new corners. Nine sit on the 600-cell's 120 corners, and one, the great grand stellated 120-cell, sits on the 120-cell's 600. They are the same points, joined up differently.
Ludwig Schläfli found four of them in the 1850s. He set the other six aside because their cells or corners are shaped like the two star solids where Euler's formula stops giving 2. Edmund Hess completed the list of ten in 1883.
| Name | Symbol | Where the star is | Cells | Found by |
|---|---|---|---|---|
| Small stellated 120-cell | {5/2,5,3} | Faces | 120 small stellated dodecahedra | Hess |
| Great stellated 120-cell | {5/2,3,5} | Faces | 120 great stellated dodecahedra | Schläfli |
| Great grand stellated 120-cell | {5/2,3,3} | Faces | 120 great stellated dodecahedra | Schläfli |
| Great 120-cell | {5,5/2,5} | Cells | 120 great dodecahedra | Hess |
| Great grand 120-cell | {5,5/2,3} | Cells | 120 great dodecahedra | Hess |
| Great icosahedral 120-cell | {3,5/2,5} | Cells | 120 great icosahedra | Hess |
| Icosahedral 120-cell | {3,5,5/2} | Around each edge | 120 icosahedra | Hess |
| Grand 120-cell | {5,3,5/2} | Around each edge | 120 dodecahedra | Schläfli |
| Grand 600-cell | {3,3,5/2} | Around each edge | 600 tetrahedra | Schläfli |
| Grand stellated 120-cell | {5/2,5,5/2} | Faces and edges | 120 small stellated dodecahedra | Hess |
5D and up: none
The reason none exist in 5D and up is a pattern running through every star so far:
- Every star in 3D and 4D is built on the pentagram . Look at any 3D or 4D symbol in this section and there's a 5/2 in it. The other 2D stars, like {7/2}, never make it past the flat plane.
- Pentagrams need the pentagon's family of symmetry . Only four regular shapes have it: the dodecahedron and icosahedron in 3D, and the 120-cell and 600-cell in 4D. Every star solid and star polychoron sits on the corners of one of these four.
- That family ends at 4D. There is no 5D dodecahedron or 5D 120-cell, which is why the dodecahedron's line stops at the 120-cell. From 5D up, only the simplex, cube and orthoplex remain, and none of them is built on pentagons.
With nothing left to build a pentagram on, there are no regular star polytopes from 5D upward. The three convex families are the whole story.
Try it yourself
Hyperslice is an interactive geometry lab I built, and it's live here on my website. It comes in two parts.
02 / One dimension down
How a slice works
The same idea one dimension down: a sphere passes through a wall, and the wall sees only a circle.
Open the primer →Further watching
- Perfect Shapes in Higher Dimensions by Numberphile · YouTube
- The things you'll find in higher dimensions by Zach Star · YouTube
- All 7 Dimensions Explained in Detail (From 0D to Infinity) by Ridddle · YouTube
A note: only the first three dimensions here are spatial in the geometric sense. The video treats time as the 4th, and its higher dimensions are metaphorical and philosophical rather than geometric.
- Professor Leonard Full-length university math lectures · YouTube
- Grant Sanderson / 3Blue1Brown Animated visual explanations of math · YouTube
- Geometry lessons Grant Sanderson · 3Blue1Brown
Articles
- A Short History of Multiple Dimensions David D. Nolte · Galileo Unbound
- Platonic Solids in All Dimensions John Baez · UC Riverside