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.

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.

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

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.

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.

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.

Waving windows to the 4th dimension.

I had the same dream a week later.

My dream under the intense scrutiny of psychoanalyis.

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

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?

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.

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

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
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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! |