By Chris McMullen
This colourful, visible creation to the fourth measurement presents a transparent rationalization of the recommendations and diverse illustrations. it's written with a marginally of character that makes this an attractive learn rather than a dry math textual content. The content material is particularly available, but whilst distinct sufficient to fulfill the pursuits of complex readers. This e-book is dedicated to geometry; there aren't any religious or spiritual parts to this ebook. could you take pleasure in 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: third-dimensional Space
- Chapter four: A Fourth size 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: residing in a 4D House
- Further Reading
- About the Author
Put in your spacesuit, strap in your defense harness, swallow your anti-nausea drugs, and revel in this trip right into a fourth measurement of house! 10D, 9D, 8D, 7D, 6D, 5D, 4D, 3D, second, 1D, 0D. Blast off!
Read or Download A Visual Introduction to the Fourth Dimension (Rectangular 4D Geometry) PDF
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Additional resources for A Visual Introduction to the Fourth Dimension (Rectangular 4D Geometry)
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. There are 6 different ways that 2 of the 4 hyperplanes can intersect. In each case, the region of intersection is a plane. This is shown in the following picture. Now let's consider how planes, hyperplanes, and their intersections relate to the structure of the cube and tesseract.
When you imagine that C is in the front, you picture a cube extending up and to the right; but if you imagine that C is in the back, you picture a cube extending down and to the left (as if you're inside a room looking at a corner on the floor). The two bottom figures show how both possibilities can be constructed by folding the L-shape two different ways. Mathematically, we can graph 3D using a coordinate system. If you stand in the corner of a room and look at the corner on the floor, this can help you to visualize a 3D coordinate system, with x-, y-, and z-axes.
This is exemplified in the following figure, which shows three different sets of coordinates that result in a point lying at the same position in the sheet of paper. Although there is a little ambiguity in drawing 3D objects on 2D paper, there is even more ambiguity in drawing 4D objects on a 2D paper; and to make matters worse, we don't have experience living in 4D space. Nonetheless, we can still learn a lot about the geometry of the fourth dimension. In this book, we will approach this from a conceptual standpoint, but keep in mind that it is straightforward to work out the algebra for the fourth dimension.