Square Circle Challenge

[MESchlum]

After muttering through Godel I have decided to try something fun. Maybe this should go into Puzzles, but this is the math area...

It has been said (somewhere here ) that a square cannot be a circle (sorry, a cube cannot be a sphere).

Hence the challenge: prove (using the geometry of your choice) that a square can be a circle.

I'll start:

Consider a non-Euclidian space, to be precise, the surface of a sphere of radius R.

A Circle is a set of points all of which are the same distance from a given center C.

A Square is a quadrilateral, each side is the same size and each angle is equal.


Take a point C, and look at the circle of radius pi R / 2 and center C. This circle is a great arc of the space, and so is also a straight line by local definitions.

Consider the squares (in the non Euclidian space) that have C as a center (intersection of the diagonals). All summits of the square are the same distance from C, and so are in a circle (non-Euclidian) of center C. Of course, the sides are, as a general rule, not in the circle, since they are parts of great arcs.

Take, in particular, the square whose diagonal is pi R / 2 long. The only great arc connecting two adjacent sides is on the circle - hence this specific square is a circle.

Thus, in this non-Euclidian space, every circle of radius pi R / 2 is also a square (and a triangle, for that matter).

Neat!

Next...

 

[drkitten]

After muttering through Godel I have decided to try something fun. Maybe this should go into Puzzles, but this is the math area...

It has been said (somewhere here ) that a square cannot be a circle (sorry, a cube cannot be a sphere).

Hence the challenge: prove (using the geometry of your choice) that a square can be a circle.

Use taxicab geometry, also known as the Manhattan metric or more formally the L1 metric.

The "line segments" connecting (0,x), (x,0), (0,-x), and (-x,0) are at distance x from the origin throughout, and hence this figure is a "circle" centered on the origin. The segments are also of uniform length and all angles are congruent, thus the figure is also a square.

 

[c4ts]

A Circle is a set of points all of which are the same distance from a given center C.

I thought that was a property of a circle, that all possible points on the circle are equidistant from the center, not a definition of it. Unless you are suggesting that geometric figures are made up of points. The way I understand it, points do not take up space, so an infinite number of them isn't going to produce anything that can occupy space, but a circle can- it has circumference, diameter, radius, area- all of which can be measured. Even in non-euclidian geometry, a point is still a point.

 

[Iconoclast]

prove (using the geometry of your choice) that a square can be a circle.

I believe in 1964 Dahl proved that squares can look round.

 

[Hellbound]

Re: Re: Square Circle Challenge

I thought that was a property of a circle, that all possible points on the circle are equidistant from the center, not a definition of it. Unless you are suggesting that geometric figures are made up of points. The way I understand it, points do not take up space, so an infinite number of them isn't going to produce anything that can occupy space, but a circle can- it has circumference, diameter, radius, area- all of which can be measured. Even in non-euclidian geometry, a point is still a point.

One point: There are different kinds of infinity, and sometimes infinity times zero is more than zero. This is the basis of using Calculus to, for example, figure the area under an irregular line...you assume an infinite number of rectangles each with a zero width (lines, in other words) and sum up the area. Infinities are funny things (and the cause of much fretting in quatum calculations, I understand).