By Chris McMullen
This colourful, visible creation to the fourth measurement presents a transparent clarification of the thoughts and various illustrations. it truly is written with a marginally of character that makes this an enticing learn rather than a dry math textual content. The content material is particularly available, but while particular adequate to meet the pursuits of complicated readers. This publication is dedicated to geometry; there aren't any non secular or spiritual parts to this e-book. might you take pleasure in your trip into the interesting 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: 3-dimensional 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 safeguard harness, swallow your anti-nausea drugs, and revel in this trip right into a fourth measurement of area! 10D, 9D, 8D, 7D, 6D, 5D, 4D, 3D, 2nd, 1D, 0D. Blast off!
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Extra info for A Visual Introduction to the Fourth Dimension (Rectangular 4D Geometry)
Now let's consider how planes, hyperplanes, and their intersections relate to the structure of the cube and tesseract. The cube is bounded by 6 squares (which are planar). Two adjacent square faces intersect at a line (each edge). The 12 edges (monkey tails) are the regions where two square faces intersect. The cube is bounded by 6 square faces, which are joined together at 12 edges. Similarly, the tesseract is bounded by 8 cubes (which are hyperplanar). Two adjacent cubes intersect at a square (which is planar) in 4D.
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. So the formula for the number of squares in a tesseract (full of monkeys) is 6 times 16 divided by 4 = 24. The 6 is how many squares meet at each corner; there are 16 corners; and we divide by 4 because each square has 4 corners (so 6 x 16 counts each square 4 times; dividing by 4 corrects for over-counting).
Two adjacent square faces intersect at a line (each edge). The 12 edges (monkey tails) are the regions where two square faces intersect. The cube is bounded by 6 square faces, which are joined together at 12 edges. Similarly, the tesseract is bounded by 8 cubes (which are hyperplanar). Two adjacent cubes intersect at a square (which is planar) in 4D. ) The 24 squares are the regions where two cubes intersect. The tesseract is bounded by 8 cubes, which are joined together at 24 squares. Click here to return to the Table of Contents.