divbyzero - Network

My undergraduate honors thesis student, Megan Triplett, gave a great thesis defense yesterday on "Finite and Countable Dynamical...

My undergraduate honors thesis student, Megan Triplett, gave a great thesis defense yesterday on "Finite and Countable Dynamical Systems."

The idea: if you have a function f:X→X, then you can iterate it to get a discrete dynamical system. "Orbits" have the form, x, f(x), f(f(x))=f²(x), f³(x),... If X is a finite set, then every orbit is eventually periodic. (If f is invertible, then every orbit is periodic.)

Megan mainly was looking at examples where X was a countable set (like the natural numbers ℕ), but in which the dynamical system behaved like a finite dynamical system—basically, if every orbit enters some finite set eventually, then we know every orbit is eventually periodic.

For instance, one big collection of examples she looked at was what she called "Stewart functions" (based on a 1960 article by B.M. Stewart). Suppose we have any function on the digits f:{0,1,...,9}→ℕ (she also looked at base B examples). Define a function F:ℕ→ℕ by F(a)=Σf(aᵢ) where a=aᵣ...a₁a₀ is a written in base 10.
Examples could include the sum of powers of the digits.
F(123)=1+2+3=6
or
F(123)=1²+2²+3²=14
or
F(123)=1³+2³+3³=36
Or you could take f(d)=dᵈ (you can choose whether f(0)=0 or f(0)=1)
F(123)=1¹+2²+3³=32
It turns out that EVERY Stewart function behaves like a finite dynamical system in that every orbit is eventually periodic and there are finitely many periodic orbits.

In that last example, every number ends in a fixed point: 1 or 3435 (or 438579088 if we assume 0⁰=1). 3435=3³+4⁴+3³+5⁵ is called a Munchausen number.

She looked at many other examples and proved some nice theorems along the way. Congratulations to Megan! She'll be in math grad school at Ohio State next year.

I designed and 3D printed a couple of models of Riemann sums for multivariable functions—the monkey saddle and the sombrero function. (I...

I designed and 3D printed a couple of models of Riemann sums for multivariable functions—the monkey saddle and the sombrero function. (I also got to try out this cool filament.) The files are on Thingiverse. https://www.thingiverse.com/thing:7019158 https://www.thingiverse.com/thing:7019184

I uploaded printable files for both cylinders to Thingiverse....

I uploaded printable files for both cylinders to Thingiverse.
https://www.thingiverse.com/thing:7006245
https://www.thingiverse.com/thing:7006240

I printed another cylinder.

I printed another cylinder.

My weekend project was designing this cylinder. It took over 24 hours to print. I have a different one on the printer right now. I'll...

My weekend project was designing this cylinder. It took over 24 hours to print. I have a different one on the printer right now. I'll post a photo tomorrow.

I have a new article in Math Horizons, "Folding a...

I have a new article in Math Horizons, "Folding a Cardioid."
https://www.tandfonline.com/doi/full/10.1080/10724117.2025.2461417#d1e300

Here’s my latest. Shine a light through the hole in the cylinder and produce a heart!

Here’s my latest. Shine a light through the hole in the cylinder and produce a heart!

Proof of concept: Shining a light through a cylinder with a shape cut out produces a shape on the ground. I had to wrap the paper around a...

Proof of concept: Shining a light through a cylinder with a shape cut out produces a shape on the ground. I had to wrap the paper around a flower vase and tape some of the pieces flat to the side. Still, I think it worked out pretty well for my first shot!

For "π+ε Day."

For those of you interested, here's a summary of of what I heard (eight responses).Number of outside speakers per semester: 2, 1, 4, 0,...

For those of you interested, here's a summary of of what I heard (eight responses).

Number of outside speakers per semester: 2, 1, 4, 0, 7, 4-5, 13, 6
Honorarium: $100, $150, $150, $0, $0, $75, $125, $100
Travel expenses: most find local speakers
Meals with speakers: food is usually involved, either at the talk or at a meal afterward.

Thank you to those of you who replied!

I'm curious what other undergraduate institutions do about outside speakers for their colloquia. I'd love it if you filled out this...

I'm curious what other undergraduate institutions do about outside speakers for their colloquia. I'd love it if you filled out this survey. I'll share the results here (on Mastodon) and will email the results to anyone who leaves an email address on the form. Thanks! https://forms.office.com/r/D7ASumjnqf

New blog post: I figured out how to remap my keyboard (Mac) to make typing math easier:option-f: φ
shift-option-R: ℝshift-option-!:...

New blog post: I figured out how to remap my keyboard (Mac) to make typing math easier:
option-f: φ
shift-option-R: ℝ
shift-option-!: ≈
shift-option-V: ∉
option-[: ∩
shift-option-<: ⊆
Ex. all from my keyboard:
∫3cos(θ)sin²(θ)dθ=sin³(θ)+C
https://divisbyzero.com/2025/01/18/type-math-on-a-mac-remapping-my-keyboard/

What is the envelope of the lines from (-a,y) to (a,1/y) for all y≠0? An ellipse (acircle when a=1)! Here's an applet I made where you...

What is the envelope of the lines from (-a,y) to (a,1/y) for all y≠0? An ellipse (acircle when a=1)! Here's an applet I made where you can play with it. The red lines pass through evenly-spaced points on x=-a, and the blue lines pass through evenly-spaced points on x=a. www.geogebra.org/m/qaqh4f7j

My colleagues and I have an article in the Monthly about starting a Data Analytics major here at Dickinson. Our path was a little unique...

My colleagues and I have an article in the Monthly about starting a Data Analytics major here at Dickinson. Our path was a little unique (and everything happened very quickly), but we hope faculty members at other schools find it helpful. https://www.tandfonline.com/doi/full/10.1080/00029890.2024.2409596

Usually, you think of "magic eye" illusions as allowing you to see a 3D shape in a 2D image. With this example that I made, you...

Usually, you think of "magic eye" illusions as allowing you to see a 3D shape in a 2D image. With this example that I made, you can see a 4D shape! I tried to optimize this for phone viewing in landscape mode.

With the help of generative AI, I figured out how to take an stl file (like those you'd 3D print), import it into Blender and output as a grayscale depth map. Then, using AI-generated Python code, I created the csv depth map that Excel can read.

I just wrote a blog post describing how to make a "Magic Eye" image using Excel. I also give a sample Excel file for you to start...

I just wrote a blog post describing how to make a "Magic Eye" image using Excel. I also give a sample Excel file for you to start with. Have fun! https://divisbyzero.com/2024/11/30/make-a-magic-eye-image-using-excel/

Because botsin.space is shutting down, I had to move my Random Tiling Bot (which produces one randomly generated tiling patern per day). The...

Because botsin.space is shutting down, I had to move my Random Tiling Bot (which produces one randomly generated tiling patern per day). The new home is @RandomTiling. If you followed it before, you should now be following the new account—no action required. Thank you to @Alien_Sunset for encouraging me to move the bot and for instructions on how to do so!

@RandomTiling And a couple more

@RandomTiling And a couple more

For those of you who followed my "random tiling bot" on Twitter (before Musk shut down bots) and @RandomTiling here (which will...

For those of you who followed my "random tiling bot" on Twitter (before Musk shut down bots) and @RandomTiling here (which will soon disappear because the service is shutting down), here are some random tilings made using Excel. These are colored Excel cells (250x250), zoomed way out.

East Central State University's math department boasts an average class size of 24. The students pushed back and said that, on average,...

East Central State University's math department boasts an average class size of 24. The students pushed back and said that, on average, they have 65 students in their classes. How is this possible?

The department offers classes with 100, 20, 20, 10, 10, 5, and 5 students in each.

The department points out that the average size is (100+20+20+10+10+5+5)/7=24.

On the other hand, of the 170 students, 100 are in a class of size 100, 40 are in a class of size 20, and so on. So the average for the students is
(100•100+40•20+20•10+10•5)/170=65.

(This is why a restaurant owner can complain that business hasn't been good even though it is crowded on days that you go.)

I've been thinking about this while doing college tours with my daughter, and colleges talk about average class size.

In this 1982 article, www.jstor.org/stable/2690083, David Hemmingway shows how these two averages are related. The college's average value is always less than or equal to the students' average value, and the discrepancy is larger as the variance of the class size increases.

Here are a few more fractals made using Excel. These are randomly generated "Barnsley ferns" (with different parameter values)...

Here are a few more fractals made using Excel. These are randomly generated "Barnsley ferns" (with different parameter values) made in 250x350 spreadsheet cells. https://en.wikipedia.org/wiki/Barnsley_fern