By Chris McMullen
This colourful, visible advent to the fourth size presents a transparent rationalization of the thoughts and diverse illustrations. it really is written with a slightly of character that makes this an interesting learn rather than a dry math textual content. The content material is especially obtainable, but while distinct adequate to fulfill the pursuits of complicated readers. This publication is dedicated to geometry; there aren't any religious or spiritual parts to this booklet. could you get pleasure from your trip into the interesting global 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: move Sections of a Tesseract
- Chapter 12: dwelling in a 4D House
- Further Reading
- About the Author
Put in your spacesuit, strap in your defense harness, swallow your anti-nausea medication, and revel in this trip right into a fourth measurement of area! 10D, 9D, 8D, 7D, 6D, 5D, 4D, 3D, second, 1D, 0D. Blast off!
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Additional info for A Visual Introduction to the Fourth Dimension (Rectangular 4D Geometry)
It means 2 times itself N times: 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, etc. For example, a cube (N = 3) has 23 = 8 corners and a tesseract (N = 4) has 24 = 16 corners. Without even drawing one, we can now predict that a 5D hypercube (remember, it's filled with nuts) will have 25 = 32 corners. To see this, consider that each corner has N coordinates in N-dimensions. For example, in 3D the coordinates are (x,y,z) and in 4D they are (x,y,z,w). Each coordinate can be one of two values (0 or 1). That's why the formula for the number of corners in an N-dimensional hypercube (full of monkeys) is 2N.
For example, consider the cube below. A monkey is rotating this cube about the dotted line (it's the axis of rotation). Every part of the cube travels in a circle that rings around the axis of rotation. Viewing from left to right, you can visualize the rotation of the cube as the red square comes down in the figure below. The square shown below (not above) is perpendicular to the page at all times. A monkey rotates this square so that the left edge comes up and finishes at the right.
Here we have the 4 hyperplanes in color. In this figure, you can see how they correspond to the cubes bounding the tesseract. The sides of the hyperplanes come in 6 colors – one for each of the 6 planes. The many resulting colors have to do with transparency (you can see "through" the faces partially, so the colors that you see are a combination of 2 or more planes). In the next figure, each hyperplane is a solid color. The colors are much more straightforward this time, if not quite as pretty.