A diversion: On square circles

25 November, 2007 at 9:18 am (mathematics) (, , , )

This is a semiformal proof. A formal one would be hard to understand without pictures and would require me checking out the English translations of a number of words, which I am not inclined to do.

A square is defined as an ordered set of four points, no three of which are on the same line, and further that the angles thus created are all right (there is a bit more, but it is not too relevant). Circle is defined as a set of points that are from given distance (radius) from a given point. Because a square does not exist in hyperbolic geometry, it is sufficient to think about Euclidean geometry. Assume that there is a square circle. Let A and B be two points of the circle that are part of the same edge (i.e. that edge does not go through the circle’s center). Let C be a point that is between A and B. Distance from The center of the circle to C is lesser than the distance to A or B, so it is also lesser than the radius. Accordin to one axiom I am a bit too lazy to check out, it is possible to find D so that D is behind C when observed from the circle’s center and D’s distance from the center is the circle’s radius. Hence D is part of the circle. D is not anyof the original points that defined the circle, because if it was either of the unnamed ones, the definition of rectangle would get messy, and if it were A or B, C would also be A or B, which contradicts the choice of C. Hence, there are no square circles. Qued est demonstratum.

This diversion due to another discussion.

6 Comments

  1. chattydm said,

    Damn man, tackling metaphysics on the web is not easy and, I believe, seldom rewarding. I’ve seen the original discussion and will not comment.

    Although I do respect you tackling this, I shall limit my musings on RPGs and all things I find cool :).

    I do have one question, what is your 1st language? German? Mine’s French and I’ve found that discoursing in another language forces one to carefully chose words and simplify arguments…. While this improves legibility it often occludes the finer points of debate one would make with our first language.

    Welcome to the RPG blogosphere and thanks for making me part of your community. I’ll leave you with theory and I’ll focus on the craft! :)

  2. Tommi said,

    Thanks for the, ahem, encouragement and the welcome. I’m from Finland, myself, and also interested in the crafty aspects of roleplaying.

    Though I’m not the greatest fan of D&D, there is still is good stuff on your site/blog. Keep it up.

  3. Asmor said,

    To be fair, you used a definition of a square which seems rather specific to Euclidean geometry. There are squares in hyperbolic geometry, they just have a different definition. So really what you’ve proven is that no Euclidean square is a circle in Euclidean geometry.

    See Taxicab Geometry, where every circle is a square.

  4. Tommi said,

    Granted, I am using the definitions I have been taught. I do think that there are no square circles in hyperbolic geometry, either. The others I have little experience with, but will check out the link. Thanks for the correction.

  5. Tommi said,

    Having thought a bit further: Technically, a quadrilateral was defined to us as a set of four ordered points (A, B, C, D), no three of which are on same line, and so that AB and CD don’t interject and AD and BC don’t intersect.

    A discrete Taxicab geometry does have circles consisting of four points, no three of which are on the same line, but they are not ordered. Hence, technically, they don’t qualify as quadrilaterals and therefore as squares according to the definition I was taught in my geometry course.

  6. melcartera said,

    Hi, tommi.

    Just dropped by to update you that I’ve got a new post on my sife, “Are the New Testament documents reliable?” Just thought you might be interested, since you did post a comment on the post which started the party :) . If this is too much of a diversion, then I apologize and please delete this comment :( .

    Anyway, thanks very much for your earlier visits, and I do hope you’ll drop by again.

    God bless you!

    Mel

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