What Is A Equilateral Polygon: Fill & Download for Free

What is the definition of a square? According to Euclid a square is a quadrilateral figure that which is both equilateral and right-angled.Unless you change the definition, a square is a quadrilateral, a quadrilateral is a four-sided figure, and four does not equal five, so there are no five-sided squares.Let’s make up a definition for something like a square, but allow it to be a polygon with any number of sides. Call it a “quasi-square”. A quasi-square is a polygon that which is both equilateral and right-angled.Unfortunately, in Euclidean geometry, you can show any polygon that is both equilateral and right-angles has four sides, so that didn’t get us anything.Let’s leave Euclidean geometry and go to hyperbolic geometry. A hyperbolic quasi-square is a polygon in a hyperbolic plane that which is both equilateral and right-angled. Are there any such things?Yes, here’s one displayed the Poincaré disk model of the hyperbolic plane. In fact, you can tile the plane with them.The figure is actually a hyperbolic regular pentagon. It’s got five sides and five angles, each angle being 90 degrees.Here’s another. It’s a hyperbolic regular octagon. It’s got eight sides and eight angles, each angle being 90 degrees.You could call these hyperbolic quasi-squares, but they already have good names, so I wouldn’t use that term.In summary, there are no squares with five sides. There are other things that share some of the attributes that squares have, but they’re not squares.

The On-Line Encyclopedia of Integer Sequences® has the list of these numbers at A006561 - OEIS.Starting with a square they have 1, 5, 13, 35, 49, 126, 161, 330, 301, 715, 757, 1365, 1377, 2380, 1837, 3876, 3841, 5985, 5941, 8855, 7297, 12650, 12481, 17550, 17249, 23751, 16801, 31465, 30913, 40920, 40257, 52360, 46981, 66045, 64981, 82251, 80881, 101270. Note that the sequence doesn't always increase. For example, there are 330 intersections in the 11-gon but only 301 in the 12-gon.Bjorn Poonen & Michael Rubinstein wrote an article "The number of intersection points made by the diagonals of a regular polygon" in 1997. The analysis of this problem is complicated. Here's Figure 1 from their paper(You can open the image in a separate window to see more detail.)

Pick’s Theorem:Amazingly, the area of any lattice polygon P, i.e. a polygon whose corners have integer coordinates, can be calculated EXACTLY by the formula:[math]I(P)+B(P)/2–1[/math]where [math]I(P)[/math] is the number of lattice points inside the polygon and [math]B(P)[/math] is the number of lattice points on the boundary.Example 1: The area of the unit square with corners (0,0), (1,0), (1,1), (0,1). [math]I(P)[/math] is 0 since there are no coordinate points inside. [math]B(P)[/math] is 4, one for each corner. So [math]I(P)+B(P)/2–1=0+(4/2)–1=1[/math] which works!Example 2: The area of the polygon in the figure is [math]5+(9/2)-1=8.5[/math].As a consequence, every lattice polygon, which can have n sides of irrational length, has an area which is a multiple of 1/2.Also, it is impossible to draw an equilateral triangle with integer coordinates. The area of an equilateral is [math]A=\frac{\sqrt{3}}{4}e^2 [/math][math][/math][math] [/math][math][/math][math][/math]where [math]e[/math] is the length of the side. The side is the hypothenuse of some right triangle, so [math]e^2[/math] is an integer, so the area is always irrational.