Most of us are familiar with the idea of flatlanders (thanks to Carl Sagan), 2-dimensional flat creatures that live on a plane, or a surface. A flatlander can never see a 3rd dimension but he can deal with it mathematically. A flatlander can certainly understand points, lines, circles and all other 2D geometrical objects. A flatlander needs 2 piece of information to identify any point in his universe: X and Y. He should have no problem in imagining line-lander creatures who need one piece of information to identify the positions in their universe: X. He can think that his flatland is made of infinite number of line-lands. To make a flatland you have to take a line-land and drag it in a direction orthogonal to it.

Flatlander knows that any position in line-land can be described by:

$latex Position_{Line} = (X)$

And any position in flatland can be described by:

$latex Position_{Flat} = (X , Y)$

Or

$latex Position_{Flat} = Position_{Line} , Y$

Y is a dimension that is unknown to line-landers and is the direction that we dragged the line-land to create a flatland.

We as 3-dimensional creatures can think that our 3D world is made of a flatland, dragged in a direction orthogonal to it. Any position in our world can be described by:

$latex Position_{Volume} = (X, Y, Z)$

Or

$latex Position_{Volume} = Position_{Flat}, Z$

Z is the new direction that is orthogonal to flatland.

You already got the picture, a 4-dimensional creature can drag our volume-land in a direction orthogonal to it to create his 4D world and so on.

$latex Position_{4D} = Position_{Volume}, W$ (W is the new dimension).

In other words:

A line-lander has no idea of a flatland. The position is only X for him. He describes a 0-land by X.

A flatlander describes a line-land by $latex Y = (X)$, a set of $latex X$s.

A volume-lander describes a flatland by $latex Z = (X, Y)$, a set of $latex (X, Y)$s

A 4D-lander describes a volume-land by $latex W = (X, Y, Z)$, a set of $latex (X, Y, Z)$s.

From elementary mathematics you must remember that a set of objects could be described by a function. For example if $latex F$ is a function defined on real numbers and for each real number that it receives it returns another real number then:

$latex y = F(x)$

Is the description of a line-land from a flatlander's point of view.

Other kinds of line-lands include:

$latex y = x$

$latex y = 4 x + 5.12$

$latex y = x^2$

$latex y = \sin(x)$

$latex y = 2(\sin(x) + \cos(x))$

And so on.

The line-lander only knows about X. If somebody tells him that his line-land is a part of a flat-land he doesn’t immediately find out what F function describes his line-land the flatland. But F is easily known to the flatlanders who are studying the line-land. Generally F describes the shape of the universe in a higher spatial dimension.

Let's talk about an interesting example. Assume that the line-land world as it is seen by our flatlanders is a circle. It means that $latex F = \pm \sqrt{r^2-x^2-y^2} $. The poor line-lander has no idea about the 2-dimensional shape of his world but he can find it out.

The line-lander finds out that his world is indeed bounded if he starts walking toward a direction and reaches the starting point.

A sphere is a 3-dimensional circle, F for a sphere is:

$latex F = \pm \sqrt{r^2-x^2-y^2-z^2} $

Let's rewrite these two equations in a more usual form:

1D Circle: $latex x^2+y^2 - a^2 = 0$

2D Sphere: $latex x^2 + y^2 + z^2 - a^2 = 0$

Look at the pattern:

0D | A Pair of Points | $latex x^2 - a^2 = 0$ | $latex \omega_1 := \left (x^2 - a^2 \right )$ |

1D | Circle | $latex (x^2 - a^2) + y^2 = 0$ | $latex \omega_1 + y^2 = 0, \: \: \omega_2 := \left [ \omega_1 + y^2 \right ]$ |

2D | Sphere | $latex \left [(x^2 - a^2) + y^2 \right ] + z^2= 0$ | $latex \omega_2 + z^2 = 0, \: \: \omega_3 := \left \{ \left [ \left ( x^2-a^2 \right ) + y^2 \right ] + z^2 \right \}$ |

3D | 3-Sphere | $latex \left \{ \left [(x^2 - a^2) + y^2 \right ] + z^2 \right \} + w^2 = 0$ | $latex \omega_3 + w^2 = 0$ |

ND | N-Sphere | $latex \sum {X_{n^2}} - a^2, \: \: n = 0 \: to \: N$ | $latex \omega_{N-1} + X_N^2 = 0$ |

The surface of a sphere could be described by many circles. The smallest circle at the north pole is a point, the radius of the circles grows as we reach the equator and then again shrinks back to 0 at south pole. This is best described in this form of spherical coordinates:

$latex x_1 =a\cos(\phi_1)$ is a circle. A sphere is a collection of circles stacked on each other. The circles at north and south pole have 0 radius and the radius of the circle at equator is maximum (a). We can think that the radius of these circles change by: $latex r = a\cos(\phi_2)$. This $latex \phi_2$ is orthogonal to $latex \phi_1$ (and is in the new dimension).

So $latex x_2 =a\cos(\phi_2)\sin(\phi_1)$ describes a sphere.

In the same sense you can think that a circle is made of many pairs of points. At the top of the circle the distance between the pairs is 0, it reaches a maximum in equator and again 0 in the bottom.

Now we can extend this model to higher dimension spheres. Take a 3D sphere with radius 0, increase the radius to a maximum and then shrink it back to 0; you have a 3-sphere.

$latex x_3 = a\cos(\phi_3)sin(\phi_2)\sin(\phi_1)$ is a 3-sphere (a 4 dimensional sphere)

This way you can make other higher dimensional shapes. Note that here we only talked about spatial dimensions.

For example a stack of lines in Y-direction makes a square, a stack of flat squares in Z direction makes a cube and a stack of cubes in W direction makes a tesseract.

Take a look at the projection of a tesseract (4D cube) in 3D space here.

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