Integral Calculus 2

This object illustrates the disk and shell methods for finding volumes of rotation using integration. Take the region bounded by the graph of the function f(x) = -x^2 - x + 8 and the x-axis from x=0 to x=2:

and rotate it about the y-axis. In the middle is the resulting solid of revolution 

On one side is the approximation of the volume using four disks, and on the other side is the approximation of the volume using four shells. To make it clear which is which, I have shown the same model with the pieces taken apart.

 This model is "Volumes of Hanoi" designed by mathgrrl.

Integral Calculus

I am preparing some integral calculus prints. The first is the solid obtained by revolving the region between the graph of y=4-x^2 and the x-axis on [0,2] around the y-axis, and the model on the right is an approximation of that solid using eight shells, designed by mathgrl.

Here is a view of all the calculus volumes collection. 

Four volumes

Pencil for scale

Tactile graphs for a blind math student

The last few weeks, I have been working with a colleague producing raised graphs for a blind student who is taking a math class, but who has never really been taught the details of plotting.  It has been a very interesting experience learning a few details of how to design such graphs.  So far the student has made excellent progress in understanding, since nobody has ever even tried to show this to him  before! The 3D printer makes for an excellent tool, since it allows relatively quick  of graphs tailored for the class, the student, and the situation. After the first few, it has taken 15-20 minutes to prepare a graph for printing, and around one hour to print.

Below the figures of the tactile plots, I have given the workflow and the details of what has worked and what has not. I'd be very interested to hear from anyone who tries this out!

x^2 compared to x^4

ln(x) compared to exp(x)

The top graph is a  polynomial and the bottom is 1/x

x^3 versus x^(1/3) and sin(x) on two different domains

Technical details: In all cases, my general workflow was as follows:

• Step 1. Produce  two JPG figures using Matlab: (1) A grid. I used linewidth 4, with a larger size dot at the origin for reference.  (2) A graph of a function. (Note that you could use any method at all of coming up with a graph and follow the next set of steps still.) 
• Step 2. Convert figures to vector graphics SVG format using the free program Inkscape. 

Step 1 Grid



Step 1 Graph of a function

• Step 3. Open the free program Tinkercad, and add a box approximately 80x80x1 mm.

Step 3 Box in Tinkercad around 80x80x1mm

• Step 4.  Import the SVG file of the grid into Tinkercad. It needs to fit on the box from the previous step. (In my case this involved shrinking to 8%.) It should be 2-3mm high so that it can be distinguished from the background.

Step 4 add the grid to Tinkercad

• Step 5. Import the SVG file of the function into Tinkercad. It needs to be exactly the same size as the grid so it matches up, so if you shrink one to 8%, make sure to shrink the other to 8%. However, for the graph, make it 1-1.5mm taller than the grid. At first I made it too tall, and the graph and the grid couldn't be felt simultaneously. 1mm sound small, but think about the thickness of Braille, and it won't seem so mind boggling.

Step 5 Add the graph to Tinkercad but make it 1-1.5mm taller than the grid. 

• Step 6.  Download for printing, using  file type STL.



Step 6 download for 3D printing

• Step 7. Print on printer, with a raft but without support. It takes around 1 hour to print each. 

Tiling the plane

 

While at GMU, Tyler White worked with me on his senior thesis. He then went on to get a PhD at George Washington University on mathematical questions for certain tilings in the plane. He asked if I could print a set of these three different tiles that appear in his PhD thesis. It looks really cool because though it is two-dimensional, it is a projection of a three-dimensional object, so gives the impression of depth. In fact, this picture above is so misleadingly three-dimensional looking that I had to print another one with my hand in it to make it clear that the shapes really are flat. 

The way I arranged it is boring from a mathematical standpoint, since it is completely uniform. In his thesis, he studied aperiodic tilings using this set of tiles. I've asked him to explain the details. Here's a portion of an aperiodic arrangement: 

Here's the full figure from the PhD thesis: 

Here is Tyler White's full explanation of what this represents:

This tiling is part of an infinite tiling of the plane.  This tiling was originally constructed by Richard Kenyon in his 1996 paper called “The Construction of Self-Similar Tilings” published in the journal of Geometric and Functional Analysis.  This tiling is aperiodic in the sense that if a fixed reference point is chosen, and then the tiling is shifted in any direction, it is impossible to have the exact same tiling you started with.  Kenyon’s goal in the paper was to demonstrate the construction of a self-similar tiling (this tiling is only pseudo self-similar) by using a generalized substitution.  Though the tiling is aperiodic, it is generated in an algorithmic manner by a tiling substitution.  As a result, the tiling as a certain amount of regularity to it.  In fact, for any pattern found in the tiling, there is always another copy of the pattern within a fixed finite distinct (the distance you would have to look depends on the size of the pattern you are looking for), this property is known as bounded gaps.  Former George Mason University undergraduate Mathematics major Tyler White in his 2012 Ph.D. dissertation in Mathematics at George Washington University under the direction of E. Arthur Robinson, Jr. showed that under certain conditions (the shown tiling satisfies these conditions) the tiling is also topologically mixing, which is a stronger type of regularity conditions than the bounded gap property mentioned before.

Coloring Theorems

Yesterday I gave a talk for middle school students at the GMU Math Circle showing that all the countries of the world (or any other round world) can be colored with only four colors (the rule being that no neighboring countries can be the same color.)

Actually, I didn't prove the four color theorem, because as  of yet, nobody has proved the whole thing without computer assistance. 

However I did prove (using only middle school math mind you) that the number of colors needed to color any flat map is not more than five. 

Amazingly, the answer is completely different on Planet Dunkin

on which maps can require at most SEVEN colors. and to illustrate this, I printed out this model of the torus with seven countries, all of which touch each other. 

Torus model  http://www.thingiverse.com/thing:721430 by BonyJordan. (Scaled up for classroom use. Original is half this size.)

Note this map coloring problem has an interesting history,  which I include from my talk (adapted for web attention span,  meaning I removed 90% of it):

Scanning with an iPhone

I tried out   the iPhone app 123D Catch for 3D scanning. You take many photos of an object from different angles. It uses the phone's accelerometer to know which relative angle you are located at. Here is the scanned image of a banana.

After a fair amount of cleanup of the data in Meshlab,  here is the corresponding 3D printed banana. 

Not as tasty as the original, but regardless a very impressive job. Unfortunately, my experiments with human heads are so far not nearly as successful. (I have yet to scan one that's printable.) 

Frog Innards

Here's a way to practice before the actual dissection. Practical? I don't know. Seems like it might help to learn the parts prior to having to take them out.