Symmetry, Shape and Space. An Introduction to Mathematics by L. Christine Kinsey

By L. Christine Kinsey

This article is acceptable for introductory scholars, probably in courses corresponding to schooling, artwork and structure. The textual content includes a few conventional fabric from geometry in addition to extra cutting edge themes. through the textual content, the authors position powerful emphasis on pedagogy, hands-on version construction, a guided discovery approach to studying, and so forth. a lot of the cloth is written in one of these approach that it may be utilized in the school room for enrichment initiatives, by means of potential arithmetic academics.

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Example text

This knot appears in the Book of Kells. 2. CELTIC KNOTS ᭟ Exercise 16. Draw a knot with the following grid: Why We Can Always Alternate Over- and Underpasses To return to the unresolved question of why it is always possible to alternate over- and underpasses, let us consider any closed path (so that it ends at the point where it began) that crosses itself but never has a triple crossing, where three strands come together at a point. The path with the crossings divides the plane into regions: one unbounded region surrounding the scribble, and several others contained inside.

In Exercise 7 you figured out a formula for the sum of the angles of an n-sided polygon. A regular n-sided polygon will, of course, have n equal angles. Give the formula for one vertex angle of a regular n-sided polygon. ᭤ Exercise 9. If a polygon is regular and one of the vertex angles is 165°, how many sides must it have? In general, a regular polygon can be surrounded by a circle so that the vertices are equally spaced points around the circle. For example, a regular hexagon looks like this: α β If we draw lines from the center to each of the vertices, the hexagon divides the circle into 6 pieces.

What is the vertex angle? A tiling, or tesselation, is a way of filling up the plane with repetitions of a basic tile meeting edge to edge. It is known that there are only three regular tilings: ones that use only one type of tile and that tile is a regular polygon. In making any tiling of the plane, we must make sure that the angles around any vertex add up to 360°. One such tiling by regular polygons is a pattern of square tiles, as shown on the next page. Each corner of a square contributes 90°, and four corners meet at each vertex to sum to 360°.

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