Math Related Drawings

[Picture of a knotted surface] My primary research area is geometric topology. Lately I have been looking at knotted surfaces in 4-dimensional space. Here is a picture of a knotted surface. This is constructed from Fox's example 10 in the article A Quick Trip Through Knot Theory. You can make the image bigger by clicking on it. A similar picture appears on the cover of my book Knotted Surfaces and Their Diagrams with Masahico Saito

I have been playing with symmetry a little.

Let us call this six.


Five Seven.

From the fifth dimension. I am sorry for the extreme file size here.

The Crosscap. This is a generic version of the standard cross cap with associated movie.

The cone on the cube is the square of a triangle. I learned of this from Dylan Thurston. It is a geometric interpretation of the fact that 1*1*1 + 2*2*2+ ... + n*n*n = (n*n)*((n+1)(n+1))/4.

A few sketches that I drew while in Korea. These are more representational than usual.

Spinning 1 The imagery associated to spinning the braid s1.s1.s1.s1

Spinning 2 The braid picture of the spun trefoil, after Kamada.

A knotted 3-manifold in 5-space.

The Rhombic Dodecahedron As the projection of a hypercube.

A knotted 3-manifold in 5-space alternate view.

A knotted 3-sphere in 5-space draft version.

Paper Constructing a 2-fold branched covering.

Coin Flip: Simple branch points.

Work inspired by recent museum visit

And He Built a Crooked House This one goes to eleven!

Works for Mathematical Art in Ghent

Counting laps while swimming. The title says it all: I use eggs as markers for progress.

Improvisations on a tiling of a hexagon based on a triangle times triangle.

Pdf's together book of pdfs.

the model for the next two pieces.

jpeg version.

Simplex improvisation.

Simplex improvisation-jpg.

The Mask.

The Mask-jpg.

First Ping-Pong Table.

First Ping-Pong Table-jpg.

Second Ping-Pong Table.

Second Ping-Pong Table-jpg.

Ping-Pong Super-Hero.

Ping-Pong Super-Hero-jpg.

Ping-Pong Mandala.

Ping-Pong Mandala-jpg.

Best Ping-Pong #1.

Best Ping-Pong #1-jpg.

Best Ping-Pong #2.

Best Ping-Pong #2-jpg.

Ping-Pong together.

Ping-Pong together-jpg.

Red Ping-Pong.

Red Ping-Pong-jpg.

Five Fold Symmetry

Five Fold Symmetry-jpg.

Works from 2010

improvisation #1 playing with a weird projection of the triangle times triangle

improvisation #2 again.

improvisation #3 and again.

improvisation #4 and once more.

tiles 1: I was doing a calculation on a nine-element strict 2-quandle that I encoded here.

tiles 2: Same calculation, a little improvisation.

tiles 3: Same calculation, a little more improvisation.

tiles 4: Same calculation, a little more improvisation.

tiles 5: Same calculation, a little more improvisation.

Works from 2009

Velvet Elvis # 1. A late night improvisation on the double point set of the sphere eversion near the quadruple point.

One Quarter An improvisation on the 4 copies of the triangle times the triangle that fill a hypercube.

Windows to the 4th dimension.

Waving windows to the 4th dimension.

Triangle 6 after a dream.

I had the same dream a week later.

riffing on the dream.

My dream under the intense scrutiny of psychoanalyis.

Cirque de Soleil series: # 1, # 2, # 3, # 4, #5, and the contortionists: #6.

Triangle 6 version 3 a.

Triangle 6 version 3b 2.

Triangle 6 version 3b.

Triangle 6 version 7.

Triangle 6 version 1.

Triangle 3 version 4.

Triangle 3 version 4b1.

Triangle 3 version 4b1a.

Triangle 3 version 4b1c.

Triangle 3 version 4b2.

Triangle 3 v2 c The ocean of fish in the net is how one of the above should look. The computer needs to free its mind, No?

Triangle 3 v2 b Corrected version.

Triangle 3 v2 d Corrected version.

Triangle # 4 A work developed for an up coming math art conference.

An Ocean in New Mexico An improvisation on the same data. Appropriate for a Doctor's office, no?

Triangle # 3 version 1a.

Triangle # 3 version 1b.

My computer is taking LSD. File transfer problems for Triangle 3 version 2a .

My computer is still tripping. File transfer problems for Triangle 3 version 2b .

Triangle 3 v2 d1 Maybe the computer is starting to come down.

The main idea of the geometric topology of knotted surfaces is that surfaces in 4-space when they are projected to 3-space have self-intersection. To learn more read the book Knotted Surfaces and Their Diagrams Sometimes when you take an intersecting surface in 3-space, it won't lift to 4-space.

Boy Surface (illustrated here with a disk cut away) gives an immersion of a projective plane that has one triple point. This surface does not lift to an embedded surface in 4-space. The double cover of it is the surface that does not lift to an embedding in 4-space. This is illustrated next.

An immersed sphere in 3-space that doesn't lift to four dimensions is illustrated here. See if you can see that this represents a sphere by cutting and regluing the pieces along the double curves.

Koschorke's example is a surface formed from the connected sum of 3 Mobius bands that has one triple point. It has a double cover that does lift to 4-space even though it does lift to 4-space.

A knotted surface in 4-dimensional space is illustrated on the page The Seifert Algorithm.

Some Art


Three peices that I did in an effort to understand the cartesean product of a pair of triangles. Prints of these can be ordered from me.

Etude3a. Etude4a. Etude5.

Seminar sketches

Boy's surface in movie form and the 3-sphere covering the quaternions.

Boy's Surface. quaternions.

trefoil. The handle structure in the trefoil complement (rough sketch)

Studies in Binomial and Multinomial

These are some completed things. There are some more in the works that help illustrate Pascal's recursion, and the multinomial recursion from a geomatric point of view.

Pascal cube. Observe that the 1,4,6,4,1 is represented by vertex,tetrahedon, octohedron, terahedron, vertex.

Multinomial. These are the structures for the trinomial theorem.

trinomial. A short improv on the same theme.

Don't forget to write.

J. Scott Carter
Professor of Mathematics
Department of Mathematics and Statistics
ILB 325
University of South Alabama
Mobile, AL 36688-0002

(251)-460-6264 /(251)-460-7969 FAX
click here for e-mail.

The views and opinions expressed in these web page(s) are strictly those of the author. The contents of these page(s) have not been reviewed or approved by the University of South Alabama. I am not responsible for content in linked material. Come to think of it, I am not repsonsible for much!