Teaching Complex Analysis

With A Spherical Camera

MathFest 2017

Robert Jacobson, Roger Williams University

July 28, 2017

https://slides.com/robertjacobson/complex-spherical-camera/live

https://goo.gl/dFCF1m

Follow along on your own device by visiting

Tools

Hardware

Software

Course Material

Spherical Camera

Ricoh Theta S: ~$320
Ricoh Theta SC: ~$200

https://theta360.com

Spherical Camera

$190^{\huge \circ}$ Fisheye Lens

Spherical Photos

Google Street View

Spherical Photos

Google Photos

MathFest 2017

Fun Example Spherical Photos

https://photos.app.goo.gl/lTaS9VfkeRP3ogp73

https://goo.gl/A9R7EN

Spherical Photos

Facebook

Spherical Photos Video

YouTube

Spherical Photos

Theta iOS app

Equirectangular Projection

Spherical Coordinates

$\theta$ azimuth

$\varphi$ inclination

$0$

$2\pi$

$\pi$

Cubic Projection

"Cubemap", Quadrilateralized Spherical Cube, "Quadsphere"

Image Source: Prof4 (http://proj4.org/projections/qsc.html)

Software

https://github.com/rljacobson/conformal

Software

Python?

https://github.com/henryseg/spherical_image_editing

Mathematics Meets Photography, Part I: The Viewable Sphere

Mathematics Meets Photography, Part II: Conformal Is the New Normal

Mathematics Meets Photography

Math Horizons

by David Swart and Bruce Torrence

Mircea Pitici, ed. The Best Writing on Mathematics 2012, Princeton University Press, Princeton, NJ 2012.

David Swart and Bruce Torrence
Math Horizons
Vol. 19, No. 1 (September 2011), pp. 14-17

David Swart and Bruce Torrence

Math Horizons

Vol. 19, No. 2 (November 2011), pp. 24-27

Course Material

https://github.com/rljacobson/Complex-Analysis-with-Computer-Algebra

Course Material

Explorations in Complex Analysis is written for mathematics students who have encountered basic complex analysis and want to explore more advanced projects and/or research topics. It could be used as

a supplement for a standard undergraduate complex analysis course, allowing students in groups or as individuals to explore advanced topics,
a project resource for a senior capstone course for mathematics majors,
a guide for an advanced student or a small group of students to independently choose and explore an undergraduate research topic, or
a portal for the mathematically curious, a hands-on introduction to the beauties of complex analysis.

Explorations in Complex Analysis

Michael A. Brilleslyper, Michael J. Dorff, Jane M. McDougall, James S. Rolf,
Lisbeth E. Schaubroeck, Richard L. Stankewitz, and Kenneth Stephenson

MAA Press, 2012

Henry Segerman

Spherical Video

https://www.youtube.com/playlist?list=PLtO7VmK73ItYt3yUgIOTpESh7zk3ke_0I

Henry Segerman

Spherical Video

https://www.youtube.com/playlist?list=PLtO7VmK73ItYt3yUgIOTpESh7zk3ke_0I

Henry Segerman

Spherical Video

https://www.youtube.com/playlist?list=PLtO7VmK73ItYt3yUgIOTpESh7zk3ke_0I

Henry Segerman

Spherical Video

https://www.youtube.com/playlist?list=PLtO7VmK73ItYt3yUgIOTpESh7zk3ke_0I

Course Materials

Many Other Great Small Sources

Vladimir Bulatov: Conformal Models of the Hyperbolic Geometry (1, 2, 3, 4)

Droste Effect (1, 2, 3)

B. de Smit and H. W. Lenstra Jr.: The Mathematical Structure of Escher’s Print Gallery (1)

Course Materials

The Mathematical Structure of Escher's Print Gallery

B. de Smit and H. W. Lenstra Jr.

Notices of the AMS, volume 50, number 4, pages 446-451, 2007.

"The map $h(w)$ is given by the easy formula $h(w)=w^{(2\pi i + \log 256)/(2\pi i)}$."

De-Escherization

What I Did in My Class

Mini Projects

Three mini projects throughout the semester.
Worth about two homework assignments.
Many different mini project opportunities, including projects students may suggest themselves.

Group Project

Groups of 2 or 3.
A short paper using LaTeX written as if it were a section of a textbook about the topic
A physical model somehow illustrating the topic using a spherical photo (or possibly multiple photos) they took
A research poster to accompany and explain the model and the topic.
In-class lecture for the topic
Peer critiques of classmate's projects

Mini Projects

Fractals from iterative maps

Create a Mathematica notebook exploring fractals generated by iteration of simple functions of a complex variable. For example, the function $f_c(z) = z^2 + c$ starting with $z = 0$. (What happens if you fix $c$ and let $z$ vary instead?)

Droste Spirals

Using a photo you take, construct a "Droste spiral" by transforming the photo with the map $e^{\text{rotation}(\log(z))}$. See my slides for examples and a mathematical explanation: http://slides.com/robertjacobson/deck#/11. Website to make the spiral images: external link: http://www.photospiralysis.com/.

Stereographic projection of the Riemann sphere

Derive the mathematics of external link: stereographic projection of the Riemann sphere. In particular, derive the formulas mapping the point $z = x + yi$ on the plane to the corresponding point $(x, y, z)$ on the unit sphere and vice versa. Then derive the maps from the plane to the plane that correspond to rotations of the sphere.

Mini Projects

Figure 1: Crocheted Hyperbolic Surface

Group Projects

How

Mathematica or other computer software. I will help you with this part if you need it.
LaTeX (external link: Overleaf, external link: ShareLaTeX, external link: SageMath Cloud, etc.)
Physical models (globe, or something else if you discuss it with me first)
Poster
In-class presentation

Your work will involve the following tools and activities.

Group Projects

What

A short paper using LaTeX written as if it were a section of a textbook about your topic. The audience of the "textbook" is your classmates. You are welcome to review how other undergraduate textbooks presented the topic, but your paper must be your team's own writing.
A physical model somehow illustrating the topic using a spherical photo (or possibly multiple photos) you took. Feel free to be creative. Come to me if you need coding help—don't spend hours getting nowhere!
A research poster to accompany and explain the model and your topic. The poster and model will be displayed in the MNS atrium or at SASH. You might wish to incorporate your spherical photos in the poster as well. Again, feel free to be creative and to come to me for coding help.
You will give us the in-class lecture for your topic.
Peer critiques of your classmate's projects in the form of a completed peer critique worksheet I will supply.

You and your teammate will choose a topic from the list below and create the following artifacts:

Group Projects

Grading

Your project is worth 100 points broken down according to the table below.

Points	Comprised of...
10	Your peer critique of other projects
20	Your peer's critique of your project
30	In-class presentation
30	Your written paper
10	Secret sauce (i.e. my overall judgment of your work)
+10	Bonus points possible for artistic merit
Total:	100 points (but with extra credit possible)

Physical Models

"Hardware"

Physical Models

Mathematics

Riemann Sphere

Shine a laser pointer from the north pole to a point on the plane. The beam passes through a point on the sphere somewhere.

Riemann Sphere

Many Concepts Become Natural

Special cases are no longer special:

$\lim_{z\to\infty} f(z)$
"$f(\infty)$"
For a meromorphic function $f$, $\mathrm{Res}(f(z), \infty) = -\mathrm{Res}\left(\frac{1}{z^2}f\left(\frac{1}{z}\right), 0\right)$.

Riemann Sphere

Many Concepts Become Natural

Möbius Transformations:

\{\mathrm{circles}\} \cup \{\mathrm{lines}\} \to \{\text{circles}\} \cup \{\text{lines}\}

\{\mathrm{circles}\} \cup \{\mathrm{lines}\} \to \{\text{circles}\} \cup \{\text{lines}\}

$$f(z)=\frac{az+b}{cz+d}$$
$$[z_1, z_2, z_3, z_3]:=\frac{(z_1-z_2)(z_3-z_4)}{(z_1-z_4)(z_3-z_2)}\quad \mathrm{(cross\, ratio)}$$

Riemann Sphere

Many Concepts Become Natural

Residue Theorem: If $f(z)$ is a rational function and $C$ is a sufficiently large circle, then $\oint_{C} f(z)\, dz=2\pi\, i \,\mathrm{Res}(f, \infty)$.

Riemann Sphere

Many Concepts Become Natural

The Argument Principle: If $f(z)$ is a meromorphic function inside and on some closed contour $C$ that avoids any zeros and poles of $f(z)$, then $$\oint_{C} {f'(z) \over f(z)}\, dz=2\pi i (N-P)$$ where $N$ is the number of zeros and $P$ the number of poles of $f(z)$ respectively, counted with multiplicity, inside of $C$.