An interactive guide

Beyond 3D

How many directions can a thing have? Touch, drag and slide to find out what 4D and 5D really mean.

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0corners in a 5D cube
0live experiments
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Three meanings of "dimension"

People use this word in different ways. This page shows all three, and always tells you which one it is using.

MathsA dimension is one more independent direction. A shape in 4D has 4 directions at right angles.
PhysicsSpace has 3 directions and time is 1 more. Together they are "spacetime" (4D).
TheorySome ideas say there are extra tiny, curled directions. This is not proven.
Important: a screen is flat (2D). Every 4D or 5D picture here is a projection, like a shadow. The real shape cannot fit on a screen, but its shadow tells us a lot.

Maths meaning

The dimension ladder

Take a shape and drag it into a new direction. You get the next shape. Tap a step.

Drag sideways on the shape to turn it. Colours show directions: each colour is one direction.

Maths meaning

Slices: how a higher being sees us

A flat being living on a sheet of paper cannot see up. It only sees the slice where a 3D object touches its world. Move the ball through the sheet.

Maths meaning

Folding up a dimension

A cube is made of 6 flat squares. Lay them in a cross, then fold. A 4D being folds 8 cubes in the same way to make a tesseract.

Physics meaning

Time as the 4th dimension

In physics, "4D" usually means 3 space directions plus time. Here we draw only 1 space direction (left-right) and time (up). Every point is an event: where and when.

The traveller's path is a world line. Nothing can go faster than light, so every path stays inside the yellow cone. Time only goes one way, which is why time is different from the three space directions.

Theory (not proven)

Could a 5th direction hide?

Some physics ideas (Kaluza-Klein and string theory) say a tiny extra direction could be curled into a circle, too small to notice. From far away a thin tube looks like a simple line. Zoom in.

Honest note: nobody has seen an extra dimension. This is a picture of an idea, not a measured fact.

Equations

Formula lab

Every idea above has an equation. Change the sliders and watch the numbers follow the formula.

1. Counting parts of an n-cube

corners = 2n   edges = n × 2n-1
squares = C(n,2) × 2n-2   cubes = C(n,3) × 2n-3

Each new direction doubles the corners. That is why 5D already has 32 corners and 8D has 256.

2. Distance in any dimension

d = √( x12 + x22 + … + xn2 )

This is distance from the starting point (all zeros). For two points, use the differences (x1 − y1) and so on. It is Pythagoras again: each new direction adds one more square under the root.

3. Which dimension has the biggest ball?

Vn = πn/2 / Γ(n/2 + 1)   (ball of radius 1)
Vn = (2π / n) × Vn-2

Tap a bar.

Surprise: the unit ball volume grows until 5D, then shrinks. It peaks at n = 5 (about 5.26).

4. Shadow of a 4D point

x' = x × D / (D − w)

Here x = 1 and D = 3. When w is bigger, the point is closer to the viewer in the 4th direction, so its shadow looks bigger. This is why the tesseract looks like a cube inside a cube.

5. Time as a direction

Δs2 = (cΔt)2 − (Δx)2    γ = 1 / √(1 − v2/c2)

Δs2 is the spacetime distance between two events. γ says how much a fast clock runs slow compared with ours.

6. Slice of a ball

r = √( R2 − h2 )

This is the formula behind the Slices section. R is the ball's radius and h is how far its centre is from the world you live in. The same formula works for a 3D ball seen by a flat being and for a 4D ball seen by us.

Ideas for the future

What can we do with more dimensions?

Higher dimensions are not only for science fiction. Here are real uses and ideas. Each one says if it is used today or only an idea.

Used todayGPS and time

Physics treats time as a 4th direction together with space (spacetime). Satellite clocks run a little different from ground clocks, and together the effects make them gain about 38.6 microseconds per day. GPS must correct this or positions would drift by kilometres.

Used todayData and AI

A small picture of 100 x 100 pixels is a list of 10,000 numbers, one per pixel. Computers treat it as one point in a 10,000-number "data space" and measure distance with the same formula you tried above. These are not directions in physical space.

Used todayRobots

A robot arm with 6 joints is described by 6 joint angles. Engineers call all possible angle sets a "configuration space" and plan a smooth move as a path in it. The robot still moves in normal 3D space.

GrowingQuantum computers

To write down the full state of n qubits, a computer needs 2n complex numbers (amplitudes). That is how fast the description grows, not how much usable information you can read out. These numbers are not spatial directions.

Used todayScans over time

Hospitals use "4D-CT" and 4D-MRI: 3D scans repeated over time, for example to follow how organs move as a patient breathes during radiation treatment. Here the 4th direction is time, not space.

IdeaHidden dimensions

Some theories say tiny curled directions exist. If one is ever found, it could change what we know about forces. Nobody has measured one yet.

For your talk: start with the ladder (point, line, square, cube, tesseract). Then show a slice. Then say time is the 4th direction in physics. Finish with one use from this list.

Check yourself

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