I did some research of mathematical artists and found some work by a guy named Istvan Orosz. Orosz is an man who was born in Hungary in 1951, and creates paintings that are filled with illusions. I thought the never ending staircases shown in class were cool so I enjoyed seeing further application of that type of art done by this man.
From what I have read and seen, Orosz is like a more modern-day MC. Escher who had a similar style of art.
Back in the times of Ancient Greece and Rome, many religious architectures were created, letting the people of the times honor their gods and goddesses. One of the most famous ones is the Parthenon, of the Athena Parthenos, which was erected in the name and honor of the goddess Athena, whose power was of battle-strategy and wisdom (plus olives and owls!). While many people look at the building with the idea that it is perfectly straight and linear, the Parthenon is far from such.
This article here explains how the Parthenon was created using curves and column-swelling to create a gigantic, beautiful optical illusion, while also making the building structurally sound. How you may ask? As shown above, the design decisions were made in a way that, from a distance, everything looks just as we see it – straight and linear. If it really was what we think it is, the roof would sag in the middle. and the columns would look a bit awkward. The roof is actually a dome, while the columns swell in their middles, and both designs working together help to support the massive building, which would otherwise collapse from its own weight since it’s made of stone. This article is definitely a good read if you appreciate architecture, or even if you like to be wowed by people of history and their cleverness!
While I was doing more research on M.C. Escher and his particular style of mathematical art, I found a number of other artists who used a similar technique. In particular, I found Antoni Gaudi and his particular form of architecture to be very interesting.
Gaudi was a Spanish architect born in the mid-19th century and created a number of different mosaics using geometry. The picture featured above is a greek pillar in which a “random tesselation” is pictured.
More often than not, Gaudi used symmetry to create his mosaics. Many of his works are still shown around the world. I’d recommend checking them out if any of Escher’s work interests you!
One of my favorite shows as a kid in the 90s was Batman: The Animated Series. A wonderful show that went all out to show why Batman is one of the most interesting (though I’d argue least effective) superheroes.
The episode “Baby-Doll” plays with the themes of illusions and mirrors and even today has great emotional weight to it. The episode concerns a washed-up actress who suffers from hypopituitarism and thus has not been able to grow up in size despite being a “full-grown” adult.
The actress is jealous of her former co-stars success and decides to kidnap them, when Batman comes in to rescue them, with the help of Robin. While Robin is trying to get the co-stars out from a trap, Batman is led into a fun house with mirrors and illusions.
Below we have a particularly effective emotional scene that says a lot about Baby-Doll as a character without saying anything at all. And it’s all done through the use of mirrors and illusion!
Unlike most people, I think TED-Ed videos are really fun to watch. I found a TED-Ed video on the mathematics involved in sidewalk illusions. The video is easy to understand and reveals when and how artists began creating three-dimensional art on two-dimensional surfaces. It also discusses perspective drawings and the use of a point of convergence, also known as a vanishing point. Check it out:
Good Will Hunting is a movie about a young genius, played by Matt Damon. In the movie, Damon’s character figures out the solution to a problem that has alluded mathematicians for many years. The problem is find all the homeomorphically irreducible trees such that n=10.
Now this problem really that difficult? Turns out, not really. The most challenging part to most people is figuring out what the problem wants. Well we know what trees are: connected graphs using all vertices but not creating circuits. So it this case, the number of vertices would be 10 as n=10. Now then, what are homeomorphically irreducible trees? Simply put, they’re trees that are actually different(2 tress are the same if one’s difference from the other is a slightly different angle between vertices or a different rotation between the trees), and must not have any vertices with any reducible vertices(those with only 2 lines/edges going through them). So now you can do this problem, it just takes a bit of thinking. The trees at the end should look like these:
This article is talking about the 4-color theorem and coloring maps for countries. It says that when you’re coloring a map with states or areas that are not connected (such as the US—Hawaii and Alaska) that this theorem does not necessarily work because everything is not connected on one map. Africa is considered a 4-colorable map, while South Africa does not work with this theorem.
This article also talks about the difference between a computer testing the theorems and humans testing them. It is obvious that a computer will most likely be able to come up with the most efficient coloring pattern, while a human may not initially do so. It may take a human a number of times to come up with the optimal solution for coloring with the least amount of colors.
There is a certain kind of art behind the graph coloring and the four color theorem. I mean, of course it is because there is actual color being used in a math problem. Math, where usually you have to add and subtract numbers. This painting above, created by Piet Mondrian, does not exactly represent the four color theorem. There are edges touching that share the same color, so that of course would not work in the color theorem. But this painting was done in 1930, and the four color theorem wasn’t approved until 1976. When you compare the oil painting to an actual four color theorem problem, at first glance you might jus consider both of them to be pieces of abstract art. Of course once you take a closer look, you can see that math problem behind the picture on the right. “In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.” So maybe Mondrian wasn’t trying to achieve the four color theorem with his oil painting, but he was getting close. There is math everywhere, even in art. Or sometimes the math problem turns into the art.
Vertex coloring is the simplest form of labelling a graph. “When used without any qualification, a coloring of a graph is almost always a ‘proper vertex coloring,’ namely a labelling of the graph’s vertices with colors such that no two vertices sharing the same edge have the same color.”
Also, graph coloring with the most (k) colors is called ‘proper’ k-coloring. Coloring with the smallest number of colors that is needed is called chromatic number or sometimes represented as y(G). ” A graph that can be assigned a ‘proper’ k-coloring is k-colorable, and it is k-chromatic if its chromatic number is exactly k.” Vertices are also classified in same color or color classes, independent sets are formed from such classes.
“Nelson asked: What is the smallest number of colors that you’d need to color any such graph, even one formed by linking an infinite number of vertices?”
The Hadwiger-Nelson problem is a problem that deals with the four color theorem and is it really true for infinite amount of vertices. Aubrey De Grey who is behind this says that it is not possible for four but it is for five colors. When he tried around 20,000 vertices it did not work with four. When he tried even just around 1,000 vertices it didn’t work with four either.