This page is a sub-page of our page on Expandable Stories.
• Non-Euclidean Geometry
• Einstein for Flatlanders
• Einstein for Linelanders
• The Linear War between the planets Vectoria and Vectoria’
• Shift of Basis for Stories
Other related sources of information:
• 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
Flatland – the movie:
Flatland – the limit of our consciousness:
The 4th dimension explained
(Carl Sagan on YouTube):
Understanding 4D — The Tesseract (LeiosOS on YouTube):
The HyperSquare (= Triact) that explains the HyperCube (= Tesseract)
(Ambjörn Naeve on YouTube)
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.
Interactive simulation of the projection onto 2Space of a rotating 3Cube.
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.
Unwrapping a tesseract (4d cube aka hypercube)
(Vladimir Panfilov on YouTube):
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.
Interactive simulation of the projection onto 3Space of a rotating 4Cube (by Ron Avitzur).
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).
Tesseract – 6 rotations:
5D HyperCube (= Penteract):
The sixth Platonic Solid is called the 120-cell.
It consists of 120 dodecahedrons interconnected in 4Space:
120-cell rotating in 4D
(Rob Scharein on YouTube):