Using only a circle and straight lines, it's possible to create many different curves that are quite pleasing to look at and well known mathematically. Most of the curves that are going to be explored in this post are featured at this site, which has a program for generating them, and this site which explores some of the geometry used in creating these curves. I recommend exploring both of them if you are going to create any of the designs below.
I created all of these with a pencil and a ruler, or with the free computer program Geogebra. They could be created with any tool capable of making a straight line as discussed in the previous post on creating string art.
- Concentric circles:
- Concentric circles showing 6 pentagrams of different colors:
- Heart composed of lines, partial concentric circles, and sections of a cardioid:
- Pen or pencil
- Compass for drawing circles (or images of circles or regular polygons)
- Protractor for marking circles with even marks
Take a circle and mark it at even intervals. In the picture below, I marked it every 10 degrees.
Connect one mark to another mark. The amount of marks skipped determines the size of the concentric circle created.
Take the next mark and connect it to the mark ahead of the one you connected the previous one to. Continue doing this.
This could be an interesting stopping point.
Here is the completed concentric circle. It's actually a polygon with the same number of sides as there were marks on the original circle.
You can use the inside circle as a starting point for creating another circle.
These concentric circle designs are really creating star polygon,s as discussed in the posts on creating star designs on pumpkins and creating torus knots. In the image below, I took a circle with 30 marks and connected them in a design with 6 pentagrams. This works because 30 / 5=6.
I decided to fill in some of the kites created by the intersecting lines.
There are lots of ways to create ellipses, but this one is pretty fun. Start with a circle with a number of evenly spaced marks. I again picked marks every 10 degrees on this one. I connected two of the marks that were 180 degrees apart. This line will become the major axis.
Mark a point on your major axis. Making this point further from the center of the circle makes the ellipse longer and narrower.
Make a right angle on one of the marks on the circle so that one of the sides of the angle goes through the focus. Connect the line from the mark to where it intersects the circle.
Continue this process for all of the marks on the circle.
Here it is halfway done. Notice the increased resolution as it gets further from the focus.
The completed ellipse. You could improve the resolution near the focus by repeating the process using the focus on the other side.
Taking parts of two ellipses intersecting each other at right angles creates a heart shape.
This heart shape made me think of a curve in mathematics that is named after the heart. The cardioid is a very interesting curve.
Start with a circle with a number of evenly spaced marks. I picked marks every 10 degrees on this one yet again.
Starting with one mark, connect a line from it to a mark that is two marks away.
Take the next mark after your starting point and connect it to the mark that is two beyond the ending point of the last mark.
Repeat. You are basically counting by ones on the starting points, and by twos on the ending points.
It's starting to become more curved:
Continuing on the other side by still counting by ones for the starting points, and by twos for the ending points:
The completed cardioid:
You can follow this same process only by counting by threes on the ending marks or fours and you will get more sharp points. If you skip less often, you will get a more gradual spiral. The curve below was generated by counting by twos only every 4th time.
I really need to find a color scheme other than ROYGBIV.
I decided to try and design a curve that looks more like a heart by combining linear sections, concentric circle sections, and cardioid sections. I think the final design looks pretty good. To start, you again mark a circle evenly. I used 10 degrees again. I also made the marks at 90 degree intervals more visible. These will be important marks.
Draw a line between two of the marks at 90 degree intervals. This will become part of the pointy end of the heart.
Connect the next marks as if you were making the concentric circle design.
Stop when you get to the line that connects to the point 180 degrees from the starting point.
You could do the same on the other side. You might not want to do this until you are completely done. All of the marks might be a little bit confusing for the next part.
Continuing the curve will be a section of a cardioid. Now you must connect the marks by counting by ones for the starting point of the lines, and by counting by twos for the ending points.
When you reach the halfway point you should have a line that connects marks that are 180 degrees apart.
Repeat with the other side:
Here I did it on a computer using 59 marks:
I attempted to color it:
If you make any of these designs, any from the post on creating parabolic arcs, or any of your own, let us know by posting a picture or video to the corkboard. We'd love to see them. Can anyone make them using other materials?