如何手工画分形图案

How to Make a Fractal

By Jason Padgett

This is a hand drawn fractal of the structure of space time at the quantum level (Planck length/Planck Particle size frame). It shows a time lapse film of a mathematical savant, Jason Padgett, hand draw the fractal. Color was added by computer and wave equations make his drawing vibrate. The equation describes the geometric connection between the quantum energy (very small, hw) and the the energy described by Einstein's famous E=mc^2.

Jason Padgett discovered how fractals arise from relativity and limits. He is a geometric mathematical savant and lives in Washington state where he currently studies.

How to Make a Fractal By Hand

Step 1: Draw an outline of the shape you want to draw a fractal of. In this example we will use a pentagon that is sliced into 40 equal segments. To make the pentagon perfect you must first draw a circle with a compass. Now that you have the circle drawn you divide360degrees by 40 and you get nine degrees. Now, you must take a360protractor and put its center at the center of the pentagon. (The center of the pentagon is the center of the initial circle. Just look for the small 'hole' made in the paper by the compass to find the center of the pentagon). Now take a small piece of tape and use it to tape the compass in place so it will not move as you measure the angles. Now use a mechanical pencil (3mm to 5mm pencil lead works best)to make a small dot every 9 degrees.

In other words, draw a straight line from 0/360degrees to 180. From 9 degrees to 189, 18 to 198 and keep doing this until the pentagon is sliced into 40, nine degree segments.

Note that you are NOT slicing the circle into forty segments, just the pentagon. We just drew the circle at the beginning to make drawing a perfect pentagon easier. So when you are slicing the pentagon into segments you stop the lines that you are drawing when you reach the perimeter of the pentagon.

So at this point what we have is a pentagon sliced into 40 equal segments. Now we are going to turn this shape into 40 right triangles (I use right triangles because that is how energy travels, magnetism, radar, etc.)

To do this you draw a straight line from 0/360degrees (at the perimeter of the pentagon) to ninety degrees. This makes a 45, 45, 90 degree right triangle that goes from 0/360to the center to 90 degrees and back to 0/360. Now draw a straight line from 9 degrees to 99 (remember that this line starts and ends on the perimeter of the pentagon). Then draw a straight line from 18 degrees to 108, 27 to 117, 36 to 126 etc...until you have connected all of the triangles.

At this point the fractal looks like this. (See picture of one hexagon sliced into 40 right triangle).

Now what we are going to do is translate the entire shape (the entire pentagon). Remembering back to algebra we learned that you can graph any equation. So if you take a basic function say f(x)= 2x + 1 and graph it you translate the points onto a graph and make a line. If x=0 then y=1, if x=1 then y=3 etc. We are doing the same thing with fractals only we are translating the ENTIRE SHAPE instead of just a point.

Take your compass and make sure that it is set to make the exact same size circle that we started with when we made the first pentagon. Now put the compass right on the tip of the first 'peak' of the pentagon (the first peak is touching the perimeter of the first circle at 0/360degrees). Then draw another circle (it will be the exact same size as the first circle). Now repeat the process of using this circle to make a pentagon. Then slice the pentagon into 40, nine degrees segments. Then connect the segments into right triangles just like the first.

Now repeat this process only put the center of each new pentagon on the points of the original pentagon that are touching the original circle's perimeter. And there you have it, a fractal drawn from right triangles.

This is a basic version of how to draw a fractal of anything. You can change it up by slicing any shape into right triangles or any other type of triangle. Or if you want to give the appearance of depth/motion you can draw the initial pentagon (or whatever shape your using) and then when you draw the other five pentagons (or whatever shape your using) make them all smaller (by the same factor) or make them larger (by the same factor).