How Algebra Sheds Light on Things

Egbert Brieskorn‘s mathematics books contain a lot more flesh than those structured strictly by definitions, lemmas, and theorems. What might be a hindrance to those looking for a quick grasp of a theory is a treasure trove for others who relish an extra of motivation, history, and examples. Currently, I am reading his book Plane Algebraic Curves (authored together with Horst Knörrer) which, already in the introductory chapter, contains many wonderful examples such as linkages, envelopes, and the Hopf fibration. I will come back to some of these topics in later posts. Today’s post contains a quick glimpse of caustics.

Beautiful Binomials

While leafing through the book Geometric Trilogy I – An Axiomatic Approach to Geometry, I came across two nice geometric depictions that are probably widely known but which I would love to have seen back in school. The first one is the geometric illustration of the algebraic fact that $(a+b)^2 = a^2+2ab+b^2$, the quadratic case of the binomial theorem. This can already be found in Book II of Euclid’s Elements. From this, it is easy to come up with a three dimensional version of the construction giving $(a+b)^3=a^3+3a^2b+3ab^2+b^3$. Creating a good picture of this is somewhat tedious so I am very glad that I can use this opportunity to link to a wonderful blog where this and other wonderful mathematical illustrations and animations can be found: Hyrodium’s Graphical MathLand.

Euler and the Kids

Being on parental leave, somewhat naturally, gave me more time to spend in the kids’ room. After a chance encounter with Mr Pythagoras there some time ago, it was now time  for a scheduled meeting with Mr Euler. The configuration in the image to this post is known to young parents as an Oball. To mathematicians, it is known as the truncated icosahedron. An interesting fact is that because it is constructed entirely from pentagons and hexagons, it has to have exactly 12 pentagons. Let’s see how Mr Euler can convince us of this necessity. Continue reading

What’s on Your Napkin?

Either nature or my brain or both do a wonderful job of clustering related things for me so that I often feel coming along many related things within a short time. Recently, I read Stuart L. Pimm‘s wonderful The World According to Pimm: a Scientist Audits the Earth. It contains the following sentence:

« Robert Colwell, an ecologist from the University of Connecticut, and I were sitting in a bar drinking Antarctica and excitedly drawing lines on paper napkins, the preferred medium of serious scientific discourse worldwide. »

Shortly afterwards, I learned about the blog What’s on my blackboard? showing people’s blackboards (or whiteboards) which are often stunning.

Hence my question: what’s on your napkin? I dare you: grab your smartphones and share images of your napkins full of scientific sketches and/or formulas!

Neighbourly Polyhedra

In the plane, it is relatively easy to find four convex polygons which pairwise share an edge and are otherwise disjoint. Can you find five polygons in the plane with these properties? How about polyhedra? How many polyhedra can you find such that any pair of them share a face and are otherwise disjoint?
For me, this is an example where my three-dimensional imagination fails utterly. If you have never thought about this before, I suggest to try finding as many such polyhedra as possible (why not start with seven?) before reading on.

Dating

Whatever your primary association to the heading of this post may be, the main characters in this story are elephants’ teeth and stars. Inspired by Jon Kalb’s book Adventures in the Bone Trade, I will discuss two aspects of the question: how old is this place or thing?

Beauty in Patents

In an earlier post, I wondered about the hidden beauty of scientific illustrations in books which are read only by very few specialists. A similar situation is found in the context of patents. They usually come with clarifying sketches or illustrations which range from sloppy to artistic.

Not later than from the moment Richard Buckminster Fuller filed his patent on cartography in 1944, mathematical beauty had found its way into patent applications. How many hidden gems may there be in patents? At least, having patents available online makes it a lot easier to spot some.

Soap Bubbles: a Paradise for Kids and Math Nerds

Soap bubbles are simply irresistible. As is the heading of Section 4.3 of Roberto Piazza‘s book Soft Matter, so I had to borrow it for this post. In the book, you can learn about some of the wonderful physical properties of soap bubbles. For example, I was not aware that each bubble is a double shell enclosing water in between. In this way, the hydrophilic heads of the soap molecules point inside whereas the hydrophobic tails point outside. The colours are then created by interference of light reflected from the two soap films. As water flows and evaporates between the films, colours change depending on the distance between the two films.

Perspectiva Corporum Regularium

Most of the time, I enjoy how one book leads to another in a seemingly endless bibliophilic journey into the past (it is a pity that books usually cannot refer to future books). This time it is the book Shaping Space (see short review) referring to a book with very few words and many wonderful drawings of polyhedra: Wenzel Jamnitzer’s Perspectiva Corporum Regularium from 1568.