(some) Geometric Ideas

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

1. 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.
2. 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.
3. 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...
4. 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
5. 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...
6. 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

• 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

Links

• Euler’s Formula For Polyhedra
  sshttps://byjus.com/maths/eulers-formula-for-polyhedra/

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Neil Rieck
Waterloo, Ontario, Canada.