Research that attempts to use fractal mathematics to make sense of natural complex systems has been shown to yield more precise prediction capabilities than previous prediction models that relied on statistical analysis. This work, done by the “Father of Fractals,” Dr. Benoit Mandelbrot, yields a deeper level of information that greatly increases the potential for earth scientists to understand and predict natural disasters. By studying past events in terms of order and scale, scientists are able to calculate more accurate probabilities of future natural disasters. This application of data analysis is increasingly more valuable as we combat global warming and the increased frequency and intensity of these events. Better prediction allows for increased preparedness and potential evacuation which saves lives.
I have decided to focus on the math in population ecology that determines and monitors the population of species in nature. Obviously it is not possible to tag and record every single organism of each species to get a head count. Population ecologists have to use specific mathematical equations and techniques to get educated estimates of populations and study the dynamics of those populations and how they interact with their environments. This includes collecting data on how populations behave and change over time. It is critical to study why and how they interact with their environments and not just individual organisms within the population. This information is used in a multitude of ways and helps to inform and shape our decisions on how we move through the world and the impact certain factors may have.
Here is a quick run through by my favorite cram artist:
For this blog post I was interested in the potential real world applications of origami and I came across this TED talk. I recommend everyone check it out, it’s only about 16 minutes long.
Although origami had been around for a long time, it was the implementation of mathematical strategy that has led to remarkable pieces of art that boggle the mind when considering that they come from single sheets of paper with no cuts.
This is interesting in and of itself that by determining the mathematical laws that govern the limits of paper folding people were able to push the art to the limits never before considered.
Origami follows four basic laws:
- 2-colorability, you can color any crease patterns with two colors and never have the two colors meet.
- at any interior vertices M-V=+/-2 (where M=mountain folds and V=valley folds)
- when angles around a vertex are numbered, the sum of alternating angles equals a straight line
- no self-intersection at overlaps (sheet an never penetrate a fold)
By using this information, crease patterns (the underlying blueprint) can be used to create incredibly detailed structures, especially by applying the established understanding of circle packing to create flaps.
The real world application is that there are instances when a large, sheet like structure must be delivered to its destination via a route that cannot accommodate its extended size and thus requires complex packaging or folding.
This TED talk gives several examples:
- Solar array
- Jones Web telescope lens
- Eyeglass Telescope (still in experimental phase)
- Solar Sail
- Stent for opening arteries
- Airbags in cars
I thought of my own examples:
- Parachute packing
- Protein folding
I found this an exciting and surprising discovery. And to think, that I just thought origami was for calm and relaxation. The potential application for targeted drug manufacturing, protein manipulation, and other medical delivery methods is astounding. As Robert Lang says, “Origami may end up saving lives.”
When I was growing up in the late eighties, early nineties; Magic Eye books were the rage. If you crossed your eyes or smashed your face against the page and drew back slowly, a 3-D image would appear, floating in space. It was great. There was always that one friend that couldn’t ever get it, which made it even more fun.
When we were in class Tuesday, discussing wallpaper patterns, I kept thinking about Magic Eye. I had to investigate if there was any relationship between what we were learning and the secret behind creating these crazy images.
The images in the book are autostereograms. There are different types but the one that is most like what we have been talking about is the wallpaper autostereogram.
Here are a few examples:
When you stare at the image with crossed eyes or from a wall angle, 3D images are revealed. It takes some time and practice but it’s pretty mind blowing when it happens the first time. As demonstrated by my dear friend, Markice:
It was great.
So basically, and without going into the physiology of the eye and brain, the repeating images create a pattern that establishes different visual planes. When viewed from wall-end view, the brain distinguishes between these different planes and certain parts of the image stick out or push in, giving a 2D image, a 3D perspective. Try it! You can see that a few of the images I posted show several of the characteristics we have discussed including reflection, translation, and rotation.
I think that on some level I always understood that math was in the background, behind especially technical pieces of art. M.C. Escher has always been a favorite of mine and when I was little I would try to make sketches that matched his but could never figure out how he got the proportions right (never mind that my skill never graduated beyond 3rd grade blob bodies). Dr. Plante instantly captured my attention when he put Escher’s artwork on the screen. Math has always irritated me and chafes my ego, but I when he starting connecting Escher to math, I began to see it as a language to communicate placement and detail. A mathematical map that makes distortion possible (with clever shading) and tricks the eye. Escher’s work became more logical to me and I love me some logic. Some might think that looking at his work mathematically diminishes the magic, but I think it enhances it.
Hi, I am Abby Goen. I am senior in the biological sciences major. I took precalc with Dr. Plante and heard all about this awesome class that looks at math in fun ways. I had to see it for myself. So far, so good! I look forward to getting to know you all!