By Chris McMullen
This colourful, visible creation to the fourth size presents a transparent clarification of the suggestions and various illustrations. it truly is written with a marginally of character that makes this an interesting learn rather than a dry math textual content. The content material is particularly available, but whilst special sufficient to fulfill the pursuits of complicated readers. This publication is dedicated to geometry; there are not any non secular or non secular parts to this ebook. may perhaps you get pleasure from your trip into the attention-grabbing international of the fourth dimension!
- Chapter zero: what's a Dimension?
- Chapter 1: Dimensions 0 and One
- Chapter 2: the second one Dimension
- Chapter three: three-d Space
- Chapter four: A Fourth measurement of Space
- Chapter five: Tesseracts and Hypercubes
- Chapter 6: Hypercube Patterns
- Chapter 7: Planes and Hyperplanes
- Chapter eight: Tesseracts in Perspective
- Chapter nine: Rotations in 4D Space
- Chapter 10: Unfolding a Tesseract
- Chapter eleven: pass Sections of a Tesseract
- Chapter 12: residing in a 4D House
- Further Reading
- About the Author
Put in your spacesuit, strap in your security harness, swallow your anti-nausea drugs, and luxuriate in this trip right into a fourth measurement of house! 10D, 9D, 8D, 7D, 6D, 5D, 4D, 3D, 2nd, 1D, 0D. Blast off!
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Additional info for A Visual Introduction to the Fourth Dimension (Rectangular 4D Geometry)
The previous solution involved a very short pattern (1 square, then 1 cube with 3 sets of 2 square faces). This alternate solution is based on the structure of the tesseract, and may therefore seem more convincing. In a tesseract (full of monkeys), 6 squares meet at every corner (1 square lies in each of the 6 planes). This is illustrated in the figure below. If you multiply this by 16 corners, you get 6 x 16 = 96. But the answer is not 96 because we have counted each square too many times. We actually counted each square 4 times because each square connects to 4 corners.
In 4D space, there are 6 mutually orthogonal planes (remember, they're made out of monkey wallpaper): xy, yz, zw, wx, xz, and yw. The intersection between two of these planes is either a line or a point. When there is a common axis, the intersection is a line (long, straight monkey tail). For example, zw and xz intersect along the z-axis. When there isn't a common axis, the intersection is a single point – the origin, located at (0,0,0,0). This is the case for yz and wx, for example. The 6 mutually orthogonal planes of 4D space are illustrated below in black and white.
You see the left cube above as if you are looking down a hallway. Your eye is centered widthwise and heightwise. The front square appears much larger than the back square, when in fact they are the same size. The 4 angled edges are actually parallel. The point of view for the right cube above is like looking at a block with your eye above and to the right of the block. You would see only its right, top, and front sides; the left, bottom, and rear sides are hidden from view. In both of the cubes above, the angled edges are all parallel in 3D space, but look like they intersect at a point (a vanishing point) in the 2D image of the 3D object.