All the drawings of slicings in the book made us think up this exercise: Look at tiling patterns from sources such as the artwork of M.C. Escher, tile and wall decorations in the Alhambra (in Spain), and any other complex geometric artwork. A lot of people produced this kind of work at one time or another. What comes to mind right off is Islamic art, Celtic and African, in these groups you can find some pretty crazy geometric stuff. ANYWAY... Try to look at the patterns in the artwork like so:
1. Look first at the pattern as being made up of many small, simple shape parts that are not necessarily interwoven.
2. Now try looking at the patterns as composed of complex shapes, interlocked and overlapping.
3. Now try to draw or at least just imagine (you may have to do a little section only, in the interest of time) that the pattern is a 2-D slice of 3-D shapes in 3-space.
What do the shapes being sliced look like as a group? Where are there interesting color or shape interactions formed between two or more shapes?
This is a fun exercise because it works BACKWARD from the slice, instead of forward from a 3-D object to the slice. You kind-of have to play detective, but you also have a lot of freedom to decide what the shapes look like, as long as they make the slice pattern.
And we had some questions from Chapters in the book: 1. If you split a shape in the 4-D space along an axis, would the slice-half form a 3-D projection of the 4-D shape in 3-D space?
2. Is it correct to assume that slicing can only be performed in dimension 2 or higher?
3. You only actually see 2-D shapes in 3-D shapes some times. (ie a square forming the front face of a cube) Is this the same in 4 dimensions, or as you travel up dimensions does it become harder and harder to recognize shapes from a previous dimension constituting the current one?
4. Any square (or cube) in 3 or 2-dimensional space can be divided into smaller but similar proportioned shapes, is this possible in 4 dimensions?
5. Is the fifth dimension for 4-dimensional objects time?
6. We in the 3-dimensional world like to think we have 3 degrees of freedom: forward & back, left and right, and up and down. Do they have 4 degrees in the fourth dimension, or do they have more?
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