These sculptures were created in the 1970s by Morton C. Bradley and they combine the beauty of mathematics with a rich study of color theory.
What I find most puzzling about these sculptures is the amount of thought that went into the color patterns and the underlining structure of the geometric shapes. How each piece fits together is just as exciting to think about as the way the artist decided to place the colors of these shapes.
There are many examples of a colorful geometric style appearing in design. The most widely known example of this is Charis Tsevis Olympic poster illustrations. The illustrations use the same concepts of color and form but in a two dimensional space. There is math involved in the creation of these illustrations, through a program called Synthetik Studio Artist.
Carlee Hollenbeck wrote that her students were playing with this fun pattern, wondering about the number of triangles. (From Fawn Nguyen’s great site visualpatterns.org, #38.) So I made a GeoGebra sketch to help them. (On GeoGebraTube.)
Here’s what I wrote back:
So I see these in two ways, one just thinking about how many of the smallest triangle, which is not the triangular numbers.
1, 1+3, 1+3+5… 1,4,9 square numbers!
It’s interesting to separate them into right side up and upside down…
RU: 1, 3, 6, 10…
UD: 0, 1, 3, 6,…
But in the broader puzzle sense with how many triangles of each size,
1, 4+1, 9+3+1, 16+__+3+1
The blank is for how many of the 2 size. 6 point up, and 1 point down?
So maybe it’s helpful to separate for each size up and down.
2: 5 total
Digger in GeoGebraI was watching a construction site, wondering about the odd shape of the digger’s main arm. Why would it be bent? Physics and leverage? I asked a physicist and he didn’t know.
Then I saw a chart and realized it was geometry!
Play around with your own design, and compare it to the actual. On GeoGebraTube, of course. Would it be possible to make a digger that reaches too far?
Many Different Ways of Obtaining an Ellipse
In mathematics, an ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the curve. As such, it is a generalization of a circle which is a special type of an ellipse that has both focal points at the same location. The shape of an ellipse (how ‘elongated’ it is) is represented by its eccentricity which for an ellipse can be any number from 0 (the limiting case of a circle) to arbitrarily close to but less than 1.
There are many different ways of forming an ellipse. Above are a few examples!
- An animation of the Trammel of Archimides.
- An animation of Van Schooten’s Ellipse.
- An ellipse as a special case of a hypotrochoid.
- Matt Henderson’s animation of a curve surrounding two foci.
Can you think of other ways of forming an ellipse (there’s a really obvious method that isn’t listed above…)?
How to Construct the Penrose Tiles.
Read the Wikipedia page to learn about the remarkable properties of Penrose tilings.
> I especially like the deflation as you go one.
Bicorne Graph and Hat
Inspired by a Tumblr link from geometricloci to this page from the curve encyclopedia.
A proper bicorne graph has tangents at the corners that pass through the vertex. Is the actual bicorne hat a mathematical bicorne?
I don’t understand the construction of the false bicorne on this page, though! I can’t determine how the locus point is constructed.
Play with this interesting family of curves yourself at GeoGebraTube.
Brahmagupta, the Zero Man.
Made this for my math history capstone class. See also: Brahmagupta’s Theorem in GGB. This is an Archimedes/Newton/Euler level mathematician in my opinion.
rms-error: Take a mathematician, a physicist and an engineer and lock each of them in his own cell with a can of beans, but withouth a can opener. After a few days you open the doors:
In the engineer’s room: Bumps in the can, scratches on the wall, beans spilled all over the cell.
=> He threw the can randomly against the wall until it broke open. He survived.
In the physicist’s room: On the walls are scribbles and calculations, in the corner a small food stain.
=> He calculated the force, angle and trajectory needed to open the can. He survived.
In the mathematician’s room: The mathematician is dead! On the wall there is a note: “It can be proven that there exists a unique solution to open the can.”
Punchline could be punchier.
- In the mathematician’s room: On the wall there is an elegant proof of a method for opening the can. The can, however, stands unopened with a short note: “an exercise for the reader.” => The mathematician is dead.
- In the mathematician’s room: just a short note. “Already been demonstrated. Uninteresting.” => The mathematician lies dead next to the unopened can.
- In the mathematician’s room: just a short note. “3-Dimension right circular cylinder, solved by the physicist. Let C_n = S^1 x [0,1]^(n-2) be an n-dimensional cylinder… …this completes the proof by induction. QED. => The mathematician lies dead next to the unopened can.
What’s your best go?