Geometrics - This application creates Maurer rose geometric patterns using polar graphing. The geometric patterns are controlled by 3 parameters:
· Swiping left and right
· Swiping up and down
· Pressing and holding.
A "Polar Graph" is a way of representing mathematical equations. The radius of the circle is represented with the character “r”, the angle that has been traversed around the circle is represented with the character “θ” (Theta). So a circle with a radius of 1 has the equation:
r(θ) = 1.
A "Polar Rose" curve is a polar graph with the equation:
r(θ) = sin(kθ)
As you increase the k, each increment adds to the number of “petals” that the rose has. In this application, swiping to the right will increase the number of petals by increasing k by 1 for each swipe. (The petals actually increase by 2 on even increments and decrease by 1 on odd increments) Swiping back to the left will decrease the number of petals until you get to 1, which is the starting point circle. If you keep swiping to the left, the 2 becomes 1/2, then 1/3, then ¼ and so on. The roses created with these fractional values are just as beautiful, have the same petal increase feature, but are not always symmetrical.
Introduced by Peter M. Maurer in his article titled “A Rose is a Rose”.
A “Maurer Rose” is a polar rose where we skip around the figure instead of drawing it sequentially. Instead of the angle being (0,1,2,3,4,5...), we use (0,45,90,135,180,...). Swiping up increases the number of degrees we skip by 1, and swiping down decreases the skip by 1. I have the skip number set to start at an interesting place. Somewhere around 75 I think. Once you reach 360 degrees, you essentially start over (361 degrees = 1 degree). Certain degree skips may produce “uninteresting” patterns. 45, 90, 180 are some that may produce just a solitary line.
I have also added a feature to this application where the size of the rose gets ever slightly smaller for every line drawn. This produces some of the most amazing visual effects. I do not prevent the radius from getting below 0, I merely let it become a negative radius. You'll notice that asymmetrical patterns flip when they become negative. I do keep the radius from expanding forever based on the larger of the width and height of your screen.
Pressing and holding will cycle between three modes:
· A changeable Maurer rose that connects points to the previous point in the series.
· A changeable Maurer rose that draws lines connecting to an unchanging Maurer rose.
· A changeable Maurer rose that draws lines connecting to the same changeable Maurer rose but out of phase by a set number of degrees.
Another feature I added was a color fading algorithm that attempts to reach every possible color and fade between them seamlessly.
What's New in This Release:
· Forced the second pattern to start off in an "interesting" place. (Previously, it would produce what looks like just random lines. Probably because the "pattern" is so complex our human brains don't find them "interesting" because we cannot recognize the pattern)
· Set radius and radius change manually.
· Press and hold twice to see the tan(THETA) function produce more beautiful patterns.
What's New in 6.0:
· Added the ability to set the radius and the radius change manually.
· Added new geometric patterns. Press and hold twice to see the tan(THETA) function produce more beautiful patterns.