Note that having a periodic function is not solely limited to Origin + Rad * F(AngFreq * T + offset) We can do many operations to get different results. For example, we can add two of these together. Then, we'll have something like this: Origin1 + Origin2 + Rad1 * F1(AngFreq1 * T + offset1) + Rad2 * F2(AngFreq2 * T + offset2) or Origin + Rad1 * F1(AngFreq1 * T + offset1) + Rad2 * F2(AngFreq2 * T + offset2) This new periodic function, depending on how the two inner periodic functions (F1 and F2) behave with each other, will have a radius of Rad1 + Rad2. (Of course, the radii cannot be factored from this expression, so you'll just have to go with it). (These functions are similar to having the origin change by a periodic function) For example, we can add two Sine functions together. Remeber that this is a circle (radius of 40) X = 50 + 40 * Cos(0.05 * T) Y = 120 + 40 * Sin(0.05 * T) but we can add another Sine function to this circle as well. X = 50 + 40 * Cos(0.05 * T) Y = 120 + 40 * Sin(0.05 * T) + 10 * Sin(0.4 * T) So, now, in addition to the circle with radius of 40, we have a little bit of fluctuation within the Y direction. You'll see that it effectively makes a sinewave travel along the circle. The red shows the circle without the 10 * Sin(0.4 * T) The white shows the circle with the sine function added to it. As you may be able to tell, the radius of this new figure is about 50 in the vertical direction. Also, we can do X = 50 + 40 * Cos(0.05 * T) + 10 * Cos(0.4 * T) Y = 120 + 40 * Sin(0.05 * T) + 10 * Sin(0.4 * T) to get You may wonder why there are 7 loops? It does not have anything to do with the 50 and 120 being 70 apart. The seven loops come from the two different angular frequencies... the second angular frequency is 8 times the first angular frequency. That might not appear to explain the seven loops, but it does if you consider the entire circle to be a loop as well. After all, if you set the second frequency to be the same as the first, then you get a plain circle (you can see by doing some algebra) X = 50 + 40 * Cos(0.05 * T) + 10 * Cos(0.05 * T) Y = 120 + 40 * Sin(0.05 * T) + 10 * Sin(0.05 * T) is equivalent to X = 50 + 50 * Cos(0.05 * T) Y = 120 + 50 * Sin(0.05 * T) So, that would be one loop, and has eight loops: seven on the inside, surrounded by one big loop. We can add any kind of periodic functions together. The only constraint that we have with doing periodics is that all of the function calls with T have to be periodic. So, we can have something like this X = 50 + 20 * Cos(0.01 * T) + 10 * Cos(0.07 * T) + 20 * Hxgn(0.02 * T) Y = 120 + 40 * Sin(0.01 * T) + 10 * Sin(0.07 * T) + 10 * Trpz(0.002 * T) Of course it looks like scribble scratch because the periods of Hxgn and Trpz don't match Sine and Cosine's period... eww. A nicer pic can be made by taking functions with similar periods and using angular frequencies that have common factors. For example: X = 80 + 40 * Hxgn(0.01 * T) + 10 * Scln(0.05 * T) + 10 * Trpz(0.05 * T) Y = 120 + 40 * Hxgn(0.01 * T + 1) + 15 * Scln(0.05 * T + 1) It looks a little horrendous at the start, but you can see that it comes to a nice closure within three loops. Note that in addition to addition (oops, no pun intended), we can combine two periodic functions with multiplication. (The radius changes according to a periodic function) We'll have something like Origin + Radius1 * Radius2 * F1(AngFreq * T + offset) * F2(AngFreq * T + offset) The product of the two periodic functions will also be periodic, going from -1 to 1. So, we can have something like this: X = 80 + 40 * Sin(0.02 * T) * Cos(0.06 * T) Y = 120 + 40 * Cos(0.02 * T) * Cos(0.06 * T) You can clearly see the 3 prongs of the fan are due to 0.06 being 3 times 0.02. What you may not see is that changing the second pair of angular frequencies to 8 gives you eight fan blades, instead of 4 that you might expect. This is because at the halfway point, Cos(0.02 * T) is halfway finished with the circle and is -1, Cos(0.06 * T) is also halfway at -1. These two -1s multiply together to give a +1, which means that at the halfway point, the periodic function completes a lap. At the halfway point of Cos(0.02 * T), Cos(0.08 * T) has already returned to the beginning, so it is at 1, and these multiply together to give -1, so the drawing has not completed yet. Of course, you can multiply more together to get something like fX = 80 + 40 * Sin(0.02 * T) * Sin(0.06 * T) * Sin(0.1 * T) fY = 120 + 40 * Cos(0.02 * T) * Sin(0.06 * T) * Sin(0.1 * T) Where the six big prongs come from the 0.06 and the 0.02, and the four smaller prongs combining to make 10 prongs in total come from 0.1 and 0.02. Remember that the only constraint is that T must be in a periodic function. So, we have had periodic addition where "the origin change by a periodic function", and we had periodic multiplication where "the radius changes by a periodic function". Of course, we can make the final two parameters change by periodic functions... namely the angular frequency and the offset. With the angular frequency modified, we'll have something like this: Origin + Radius * F1(AngFreq1 * F2(AngFreq2 * T + offset2) + offset1) (The radius and angfreq multiplied to give the AngFreq1. AngFreq1 has to be suitably large, otherwise the function might appear to be stuck at a certain value. X = 80 + 40 * Sin(1 * Sin(0.02 * T)) Y = 120 + 40 * Cos(1 * Sin(0.02 * T)) The red circle does not have the same radius as the white circle in the picture or in the following pictures. You can see that this function never quite reaches halfway up. This is because when Sin(0.02 * T) reaches its minimum -1, the -1 goes into the Cos as a radian... which is a positive value. So, the smile will only go up a radian on each side. If you want a semicircle-shaped smile, AngFreq will have to be set to Pi / 2 (corresponds to 90 degrees), so that the circle goes up 90 degrees on each side, forming the semicircle. Setting AngFreq to Pi or greater will give you a full circle. Greater values cause the smile to swing back upon itself. Changing an internal Sine to a Cosine turns the smile into a frown. The periodics get stranger with our functions as well: X = 80 + 40 * Hxgn(1.5 * Trpz(0.02 * T)) Y = 120 + 40 * Hxgn(1.5 * Trgl(0.02 * T) + 1) Modifying the offset gives us different results as well. Some of these results can only be fully appreciated by watching the designs draw themselves live. Others, of course, draw themselves out nicely. Origin + Radius * F1(AngFreq1 * T + offset1 + F2(AngFreq2 * T + offset2)) X = 80 + 40 * Sin(0.05 * T + Sin(0.05 * T)) Y = 120 + 40 * Cos(0.05 * T + Cos(0.05 * T)) Despite the seemingly insignificant offset functions, the graph looks nothing like a circle, because when Sin(0.05 * T) reaches its maximum at 1, Sin(0.05 * T + 1) is not at the maximum, since it is offset by a radian at this point. However, when Sin(0.05 * T) reaches its maximum, Cos(0.05 * T) reaches 0, which also causes Cos(0.05 * T + Cos(0.05 * T)) to reach 0. This point corresponds with none of the four endpoints of the arrowhead, but is a point in the middle of the horizontal point of the arrowhead. There are lots more functions out there, some named after other people and some that have no name. Good luck.