Magic Triangle PuzzleDate: 07/26/2002 at 17:02:05 From: Pieter Subject: A flaw in Euclidean geometry Hello, While surfing on the Internet I encountered a site with a triangle puzzle on it. The guy that put it on his site claims he has found a flaw in Euclidean geometry. You can view the problem here: Simeon's Triangle Puzzle http://www.simeonmagic.com/triangle/triangle1.htm It looks simple, but I just can't explain it. Can you tell me where that white square came from? Thanks in advance, Pieter Date: 08/02/2002 at 14:41:07 From: Doctor Nitrogen Subject: Re: A flaw in Euclidean geometry Hi, Pieter: I suspect the reason a blank white square appears in the bottom diagram is that in the top diagram, the green right triangle at the top and the red right triangle at the bottom are not similar triangles. As a result, the area of the entire figure does not remain the same when you move the two triangles. Go back to: http://www.simeonmagic.com/triangle/triangle1.htm and examine for yourself. This refers to the top diagram: The acute angle at the bottom left of the top green triangle is tan(beta) = 2/5, and the acute angle at the bottom left of the larger red triangle at the bottom is tan(alpha) = 3/8. Put another way, the two smaller sides of the smaller green triangle are 2:5, but the ratio of the sides of the larger red one is 3:8. Clearly, 2:5 =/= 3:8, and the red and green triangles are not similar. Why should this matter? Because if the red and green right triangles were similar, you could place the green one at the bottom left and the red one at the top right, and leave no blank white square when you moved the orange figure. But the triangles are not similar, so when you place the green one in the lower position and the red one in the higher position, you decrease the total area of the larger triangle that contains the red and green triangle and the other orange figure. In fact, Doctor Ian suggests that for the two triangles to actually act as you see on that site, the big triangle that contains both of them (as well as the orange figure) would have to have a curved, concave-shaped hypotenuse in one case and a curved, convex-shaped hypotenuse in the other. I did not work out a formal proof for this; nevertheless, that's why I suspect the blank white square mysteriously appears in the bottom diagram. I hope this got you thinking on some intriguing math. - Doctor Nitrogen, The Math Forum http://mathforum.org/dr.math/ Date: 08/02/2002 at 14:43:14 From: Doctor Ian Subject: Re: A flaw in Euclidean geometry Hi Pieter, Dr. Nitrogen's analysis is correct. The key is that the two triangles aren't similar. Here's how it looks to me. In the first triangle, moving from left to right, you move at one slope (3/8), and then switch to a steeper slope (2/5). This means that the 'hypotenuse' is not actually a straight line, but in fact is slightly concave. In the second triangle, the situation is reversed: you switch from a steeper slope to a gentler one, which makes the 'hypotenuse' slightly convex. In fact, then, neither of the 'triangles' is really a triangle at all, but a quadrilateral in which one of the angles is nearly 180 degrees. The thickness of the line is used to mask the change in slope; but the difference between the convex and concave 'hypotenuses' is the area of the white square in the bottom triangle. Does that make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ For more solutions from the Web, see: Solution to Triangle Puzzle - Rebecca S. Lindsay http://www.loisterms.com/trisolution.htm Geometric Triangle Puzzle http://home.earthlink.net/~toddwolly/vision/triangle.html Magic Triangle - Remco Donders http://gene.wins.uva.nl/~donders/magic.htm - Doctor Sarah, The Math Forum http://mathforum.org/dr.math/ Date: 08/03/2002 at 03:23:06 From: Pieter Subject: Thank you (a flaw in Euclidean geometry) Thank you both for this quick reply! It seems my eyes fooled my mind. Thanks for this understandable explanation, Pieter from the Netherlands Date: 12/18/2002 at 13:31:32 From: Michael Subject: Simeon's Triangle Puzzle Answer Here is a numerical answer, easier to see for some kids: 13 * 5 / 2 = 32.5 square units. That's the apparent size of both combined (whole) triangles. But if you calculate the areas of each of the sections, again as they appear on the puzzle, they add up to 32 square units on the top one, and 33 square units on the bottom (the one with the white square in it). This confirms that the 'slightly concave, slightly convex' analysis is right, and that the hypotenuses of both triangles are not straight lines; both are arcs, with a net area of one square unit. Date: 12/18/2002 at 14:41:40 From: Doctor Schwa Subject: Re: Simeon's Triangle Puzzle Answer Thanks! Several of the answers we link to are somewhat similar to yours, but surprisingly none of them explicitly does the area calculation. I'd add some of supporting details to your calculation: Top "triangle" has: one 8x3 triangle, area 12 two "P" shaped pieces, total area 15 one 5x2 triangle, area 5 for a total of 32. Bottom picture has those same four pieces, plus the one square unit hole, for a total of 33. Thus neither of them can actually fill the 5x13 triangle exactly. The other explanations do a good job of showing that the reason they don't fit is that the "hypotenuse" is not a straight line. The two small triangles have slopes of 3/8 and 2/5, which aren't equal to the overall apparent slope of that side, 5/13. - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ |
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