Circle and Polygons: Lines of Symmetry
Date: 04/14/97 at 15:08:57 From: Anonymous Subject: Lines of symmetry Hello! How many lines of symmetry are there in a circle? This has been an on- going conversation in our class. We've asked many teachers and we have come up with 3 answers: 180, 360, and infinitely many. Thank you for your time
Date: 04/14/97 at 22:28:18 From: Doctor Steven Subject: Re: Lines of symmetry It seems you are asking the question: "How many lines can a circle be reflected about and still be self-coincident (i.e., fall back onto itself)?" The answer is infinitely many. Take any diameter of the circle and reflect the circle about that diameter and it will be self-coincident. There are an infinite number of diameters of a circle, so there is an infinite number of such lines. Notice that the circle is also self-coincident under any rotation. So there are an infinite number of symmetry rotations of the circle. A more difficult question would be to ask how many lines and rotations of symmetry a polygon has. Number the corners of a square like so: 1 _______ 2 | | | | | | |_______| 3 4 When we flip the square about a line of symmetry or rotate the square, we will call this a rigid motion, because the square maintains its shape (i.e., it doesn't get squashed or anything). A square has 4 lines of symmetry: the horizontal line, the vertical line, and the two diagonals. It also has 4 rotations: the 90 degree turn, the 180 degree turn, the 270 degree turn and the 360 degree turn. The horizontal line flip switches 1 and 3, and switches 2 and 4. The vertical line flip switches 1 and 2, and switches 3 and 4. The 1,4-diagonal line flip switches 3 and 2 and leaves both 1 and 4 fixed. The 3,2-diagonal line flip switches 1 and 4 and leaves both 3 and 2 fixed. The 360 degree (or 0 degree, however you look at it) rotation leaves everything fixed. The 90 degree rotation moves 1 to 2, 2 to 4, 4 to 3, and 3 to 1. The 180 degree rotation moves 1 to 4, and 2 to 3. The 270 degree rotation moves 1 to 3, 3 to 4, 4 to 2, and 2 to 1. Note that following any one of the rigid motions by another rigid motion gives us a different rigid motion. For example: The horizontal flip, followed by the 90 degree rotation switches 3 and 2 and leaves 1 and 4 fixed, which is the same as the 1,4-diagonal flip. I leave it to you to figure out the symmetries of a pentagon, and polygons in general. Hope this helps. -Doctor Steven, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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