# Tesseracts for Flatlanders

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Related KMR-pages:

Non-Euclidean Geometry
Dimension
Einstein for Flatlanders
Einstein for Linelanders
The Linear War between the planets Vectoria and Vectoria’
Shift of Basis for Stories

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Other related sources of information:

Tesseracts
Flatland – the movie 2008, based on the novel Flatland by Edwin A. Abbott from 1884.
Flatland – the limit of our consciousness
A Wrinkle In Time, 2018 Disney film based on the novel by Madeleine L’Engle from 1962.
How mathematicians are storytellers and numbers are the characters
Marcus du Satoy in The Guardian, 23 January 2015

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Flatland – the movie:

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Flatland – the limit of our consciousness:

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The 4th dimension explained

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Understanding 4D — The Tesseract (LeiosOS on YouTube):

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The HyperSquare (= Triact) that explains the HyperCube (= Tesseract)

This video shows:

The 3Cube as a HyperSquare (= Triact) casting its shadow onto Flatland (= 2Space).

A rotating 3Cube can be projected onto 2Space from a 3Point outside of this 2Space, which produces the “inside-out” turning motions of its shadows in 2Space.

Analogously we can regard:
The 4Cube as a HyperCube (= Tesseract) casting its shadow onto Space (= 3Space).

A rotating 4Cube can be projected onto 3Space from a 4Point outside of this 3Space, which produces the “inside-out” turning motions of its shadows in 3Space.

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Unwrapping a tesseract (4d cube aka hypercube)

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The Scan of a Tesseract in 4-dimensional space:
(Визуализации в многомерных пространствах on YouTube)

In the first part of the video shows a scan of the usual three-dimensional cube as it collapsed and expanded. In the second part, the same thing is happening with four-dimensional cube (a tesseract) in four-dimensional space. First the tesseract loses its transparency, then it again gets.

Consider the standard Cartesian coordinate system Owxyz in four-dimensional space V. Consider V a tesseract centered at the point (500, 0, 0, 0) and side 200.
Give its two-dimensional faces of color so that during the folding of tesseract matched faces had the same color.
Install a four-dimensional camera at the origin with a distance of 100 to the projected three-dimensional space U. Define the direction of the camera in the direction of w-axis. Set in U a three-dimensional camera at the point (1100*sin(2*pi*t/10), 1100*cos(2*pi*t/10), 550*sin(2*pi*t/7.1)) with the direction to the origin of the U space and with the distance to the projection plane 100 and draw a dot projected tesseract at the U of V with the centre of projection coinciding with the four-dimensional position of the camera.

Start to collapse and expand the tesseract.
This will produce what is shown in the video.

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In this simulation the four orthogonal directions of 4Space are given by x, y, u, v. The 4Cube can be interactively rotated around the xy-, xu-, xv-, yu-, yv-, and uv-planes. In 4D, the axis (= invariant subspace) of a rotation is a 2Space (= a plane).

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Tesseract – 6 rotations:

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5D HyperCube (= Penteract):

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The sixth Platonic Solid is called the 120-cell.
It consists of 120 dodecahedrons interconnected in 4Space:

120-cell rotating in 4D