Hello, welcome to the post that goes with my IGNITE talk for AMATYC 2018. The slides and extra information and links are below.

At the AMATYC national conference, we have an IGNITE event
on Friday night. Each presenter gets 20
slides that auto advance every 15 seconds.
It is fun, exciting, and most speakers, myself included, get a little
off in their timing due to the excitement, nerves, etc. Anyway, I presented this time on “Some of My
Favorite Pictures in Mathematics”

Those pictures, with comments and links, are below. We also live-streamed the event, so if you
want, you can watch it on the AMATYC Facebook Page. (The audio
and video may not be that great, as it was done “in house”) but that is what we
have!

A general disclaimer, pictures are not proofs, but they
sometimes give some real insight into a concept or problem. In addition, we mathematicians often draw
lots of pictures to help keep track of things, so I think this is an important
idea. Also, some of these are commercial
images, so I have linked to the artists or people responsible. Some of these I found cool enough to have a
copy in my office.

The slides with comments are below. Notice that the first and last slide were
just slides pointing people here so if they wanted to follow up or see more.

__Slide 2:__This image was posted by Alice Proverbio, who is a cognitive neuroscience professor at the University of Milano-Bicocca in Italy. See the following article for more information about how this static image is not really moving, even though it seems like it is. This was to drive home the point that pictures are not proofs.

The article

__Slide 3:__Simon Beck Website is an artist who makes mathematical patterns in the snow. He also uses graph theory as he makes the designs, by deciding where to back track to get to the other parts of the image and so forth. On this image, Mr. Beck is in red in the middle.

__Slide 4:__The Mandelbrot Set in mathematics is, in a sense, the basis of Chaos Theory. This is a picture of the set, but made up like an old time map. The poster version of this is hanging in my office. Web store

Slide 5: Robert J.
Lang Store does all sorts of cool mathematical origami. One of the things I like about this design is
that it is a single sheet of paper that has been folded.

__Slide 6:__I love how these images can help visualize the topology that is at work here. The coffee cup torus is from Wikipedia, but can be found in a lot of places. The Klein Bottle is from Here

__Slide 7:__some really neat books along these lines are the “Proofs Without Words” series published by the MAA. There are a number of wonderful pictures here, and I mentioned a few in my talk.

__Slide 8:__Here are two of my favorite images from the first book. I use these without the numbers in my Calculus 2 classes and ask students what sequence and/or series these are representing. I then show the images with the numbers, and we talk about how you would calculate those series. Finally, for a challenge, I ask the students to draw the series on the other shape. i.e. the sum of reciprocal powers of three in a triangle, and reciprocal powers of 4 in the triangle. Then have them defend their answers and show the area steps correspond to the partial sums.

__Slide 9:__Yes, you can do integration by parts from the product rule, but the fact that you can tie it to areas is also cool. Often, after we cover this technique in class, I just put the basic picture up, and have the students explain what the basic picture is trying to show. A good discussion is generated, especially regarding function notation.

.

__Slide 10__: Okay, so maybe a little risque, but it generates some laughs. I then have each student write a minute paper explaining the joke, but they have to keep it “G-rated”.

__Slide 11__: There are a lot of animated pictures of the Pythagorean relationship for a right triangle, but I like this one for the intermediate shapes that it shows. For my classes, I often ask them to show at each stage why is the area the same.

**Friedrich A. Lohmüller created this**

Slide 12: I stumbled across this image via Twitter, and
there are many neat things at this website.
There is even an animated version of this diagram that you can spend
time with down towards the bottom of this page:

Slide 13: These two come via Wikipedia, but they are all
over the internet. I like this as a pair
because of the very different looking shapes drawn simply by moving the circle
inside or outside. I use this to drive
home the point that details matter in mathematics.

Slide 14: If you
haven’t visited www.mathwarehouse.com,
you should. They have pages of different
gifs and a lot of other resources. The
gifs start here

Slide 15: This is a
pretty graphic for the sum of the first n integers. There are quite a lot of other really nice
resources at the website:

Slide 16: This is an Ulam spiral that is 3000 by 3000. An Ulam spiral spirals the integers in
counter clockwise direction, and colors the primes. So you can see patterns, and gaps, and it
looks cool. In this case, I mentioned
that my favorite prime is one of these dots:

Slides 17-19: My favorite prime number. Mainly because if you speak out the number, it is 8-6-7-5-3-0-9, a famous chorus from a pop song in the 1980’s. Youtube video here

Slides 17-19: My favorite prime number. Mainly because if you speak out the number, it is 8-6-7-5-3-0-9, a famous chorus from a pop song in the 1980’s. Youtube video here

It turns out that 8675311 is also prime, so “Jenny has a
twin” was the joke I finished with.