Web App for Pi Day (March 14) Lets Users "Square a Circle" -- Almost
Tulsa-based computer programmers Isystant, Inc. developed a Web app for visualizing "squaring a circle," that is, creating a square having the same area as a given circle. The visualization reveals an easy way to construct a square that is within one percent of the size of a circle, using only a straightedge and compass.
Tulsa, OK, February 16, 2016 (Newswire.com) - Every year math lovers celebrate Pi Day on 3.14, meaning March 14th. It's an opportunity to pause and marvel at the irrational, transcendental, endless, patternless, peculiarly captivating string of digits that plays a role in all sorts of unexpected places.
Note to non-mathematicians: "Irrational" doesn't mean crazy. It means "un-ratio-able" – unable to be expressed as a ratio of integers. The number 0.5 can be expressed as the ratio of 1 over 2, or ½. As for the meaning of "transcendental"…you don't want to go there.
Actually, Pi is expressed or defined as a ratio, not of numbers but of geometric figures. The ratio of a circle's circumference to its diameter equals Pi. A circle whose circumference equals Pi has a diameter of 1.
When it comes to circles, there is a famous unsolvable geometry problem – how to construct a square of equal area. A computer programmer at Isystant, Inc. designed a Web app for Pi Day that demonstrates what a square that is (almost) the same size would look like.
In doing so, he discovered an easy way for a geometry student using the traditional straightedge and compass to construct such an almost-equal square. The Web app can be found at www.isystant.com/circleandsquare.htm. It works on mobile devices as well as larger screens.
The app shows a square that contains a circle such that the circle touches the midpoint of each of the four sides. In addition, the four midpoints are connected by line segments. Those make another square that stands on one point like a diamond and has half the area of the outer square.
The outer square fully contains the circle, and the circle fully contains the inner square. Thus the size of the circle is somewhere between that of the inner and outer squares. The Isystant app allows the user to rotate the inner square while keeping its corners on the sides of the outer square. The more the inner square rotates, the larger it gets. If it kept going, it would exactly overlap the outer square once their corners coincided. Instead, the rotation stops as soon as the area of the inner square exceeds that of the circle. The user can back up or advance the rotation one unit at a time to watch the effect.
While experimenting with one-unit movements, the programmer noticed this: Bisect one side of the square, then do the same to the resulting segments until you have a segment that begins at a corner and is one-eighth of the whole side. Do that on all sides, and then connect those one-eighth segments at their non-corner ends. The area of that inner square will be 99.47% of the circle's area.
"If someone said to you, 'quick, construct a square approximately equal in area to this circle,' you'd have no idea where to begin. Me, neither," says Scott Pendleton, the programmer. "This app surprised me by revealing an easy way."
Not necessarily a useful discovery, but for Pi buffs a delightful one nonetheless!
Isystant, Inc. provides custom programming services to Tulsa-area businesses, with an emphasis on solving "the real problem" inexpensively.