extrapolations from the 1884 book Flatland (and Lineland and Spaceland) by schoolmaster Edwin Abbott Abbott

- first imagine a single point in three dimensional space (also known as 3-d space)
- what three dimensions? These are the three spatial dimensions (x, y, z) of the Cartesian coordinate system invented by René Descartes
- this point is a mental construct with zero dimensions of its own. This is easy if we are talking about a location. This is a little more difficult if we are thinking about a pencil point where the illusion is ruined by a magnifying glass.

- moving a single point in any direction of 3-d space will trace out a line (two end points; one line).
**line**is the name of a new higher dimensional item.

- moving a line (perpendicularly to the previous direction) will trace out a square (four points; four lines; one plane).
**square**(or plane) is the name of a higher dimension item. Is the plane one surface or two? Hmmm...

- moving a square (perpendicularly) will trace out a cube (eight points, twelve lines, 6 planes)
**cube**is the name of a new higher dimensional item

- moving a cube in 3-d space or 4-d space will trace out a hyper-cube
- many people have speculated that time can be considered a fourth dimension but we all know that time is not spatial. But Einstein's "theory of gravity" (also known as the theory of relativity) speaks of space-time as a real thing. Hmmm...

- Related thoughts:
- Abbott describes what a citizen of flatland might see if a 3-d sphere passed through Flatland: "a point would appear;
which would become a line; which would lengthen then contract back to a point before disappearing from view".
- Now think about about a single coil spring (like one of the two pictured pictured to the right) pushed through flatland: "a point would appear; then it would oscillate 10 times; then it would disappear".
- Pulling the spring back would make the oscillation appear to reverse direction.
- Pushing a spring wound in the opposite direction would also appear to reverse direction.
- I have often wondered if something similar could be adapted to properly explain Electromagnetism in our world without resorting to Maxwell's Laws

- Abbott describes what a citizen of flatland might see if a 3-d sphere passed through Flatland: "a point would appear;
which would become a line; which would lengthen then contract back to a point before disappearing from view".

Object Number |
Object Name | dimensions | vertices (points) |
edges (lines) |
faces (planes) |
cubes | hypercubes | dimensions | vertices (points) |
edges (lines) |
faces (plains) |
cubes | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Progression of a dot: |
Unfolded: |
||||||||||||

1 | point | 0 | 1 | - | - | - | - | - | - | - | - | - | |

2 | line | 1 | 2 | 1 | - | - | - | point | 0 | 2/1 | - | - | - |

3 | square | 2 | 4 | 4 | 1 | - | - | line segments/line | 1 | 5/2 | 4/1 | - | - |

4 | cube | 3 | 8 | 12 | 6 | 1 | - | flat cross | 2 | 14 | 19 | (6) | - |

5 | hypercube (Tesseract) | 4 | 16 ? | 32 ? | 24 ? | 8 ? | 1 | 8 cubes on a cross | 3 | 36 | 44 | 40/29 | 8 |

Other Stuff: |
|||||||||||||

3 sided pyramid (Tetrahedron) | 3 | 4 | 6 | 4 | - | - | 4 triangles | 2 | 6 | 9 | (4) | - | |

4 sided pyramid | 3 | 5 | 8 | 5 | - | - | 4 triangles, 1 square | 2 | 8 | 12 | (5) | - | |

5 sided pyramid | 3 | 6 | 10 | 6 | - | - | 5 triangles, 1 pentagon | 2 | 10 | 15 | (6) | - | |

Octahedron | 3 | 6 | 12 | 8/(9) | - | - | 8 triangles | 2 | 10 | 17 | (8) | - | |

Dodecahedron | 3 | 20 | 30 | 12 | - | - | 12 triangles | 2 | 38 | 49 | (12) | - | |

Icosahedron | 3 | 12 | 30 | 20 | - | - | 20 triangles | 2 | 22 | 31 | (20) | - |

- The Euler Formula (below) is always true for 3-d objects consisting of straight lines:

**Edges = Vertices + Faces + 2** - The yellow diagonal extrapolation suggests 8 cubes may be created when a 3-d cube is moved through a fourth (time?) dimension
**Thought experiment:**moving a cube from A to B in a 3-dimension space maps out lines between two cubes (one beginning; one ending) with connecting lines between the corners. A different visualization pictures a smaller cube inside a larger one with lines connecting the closest corners. When drawn out with pencil and paper you can see:- 16 points where lines connect. The green vertical extrapolation suggests that this thought may be correct for four dimensions.
- 32 lines ((2 x 12) + 8 new ones) but this may not be valid for 4 dimensions.
- 24 planes ((2 x 6) + 12 new ones) but this may not be valid for 4 dimensions
- all 8 cubes if you search long enough.

**Observations:**- The following formula is consistent for objects 1-5:

**Vertices = 2 ^ Dimensions** - Here's something I just noticed for objects 2-5:

**Dimensions x Vertices / Edges = 2**

- The following formula is consistent for objects 1-5:

**Euler’s Formula For Polyhedra**

sshttps://byjus.com/maths/eulers-formula-for-polyhedra/

Back to Home

Neil Rieck

Waterloo, Ontario, Canada.