THE 3,10 TORUS KNOT IN ROTATION

GROWTH & UNFURLMENT OF THE 3,10 KNOT

THE 3,10 KNOT Growing, Blossoming, Fading

THREE LOOPS IN ROTATION

THE ROUND DANCE

Science and Math
Science and Math World

THE 3,10 TORUS KNOT IN ROTATION

GROWTH & UNFURLMENT OF THE 3,10 KNOT

THE 3,10 KNOT Growing, Blossoming, Fading

THREE LOOPS IN ROTATION

THE ROUND DANCE

Much of my work

on minimal surfaces has involved branched minimal surfaces, i.e., minimal surfaces with branch points.

This page is intended to provide pictures of branch points, without

going deeply into the mathematical formulas, although the most

elementary formulas will be mentioned here.

A branch point is a point where the surface is not

parametrized in a one-to-one fashion. At that point, it doesn’t meet the

classical definition of a geometric surface, which requires that the

surface look, in the vicinity of each point, like a curved disk.

However, one is forced to consider branch points because the keep on

arising in proofs as limits of surfaces without branch points. There are

nice formulas that describe the appearance of branch points, so one can

draw pictures of them and be assured that every branch point looks like

one of these pictures. The pictures are described by two integers: the order and the index. Branch points can be interior branch points or boundary branch points. We will start with pictures of boundary branch points:

[Note: in the Safari browser, if the pictures do not rotate after they

have finished loading, use the back button and then click the link

again. They should then start rotating immediately. Other browsers do

not seem to have this problem.]

Order 2, Index 1.(fig. 1)

Order 4, Index 2. (fig. 2)

One important thing that is not obvious from the pictures is

this: the unit normal extends continuously to the branch point. That

means that all the sheets of the surface flatten out near the center and

become horizontal. In my work I am mostly concerned with a very small

neighborhood of the branch point. Then the picture is much flatter. Here

is a picture that shows a smaller neighborhood of the order 2, index 1

branch point, in which you can see how the normal tends to vertical at

the origin:

Order 2, Index 1, r=0.2 (fig. 3)

These are called boundary branch points because they

could occur at the boundary of a surface. In these examples, you see

only a neighborhood of the boundary, and the boundary in question is a

straight line. The pictures show parts of the x, y, and z axes as well. The next picture shows an interior branch point. Essentially, the first picture above is half of this picture.

Interior branch point, order 2, index 1 (fig. 4)

A branch point of order *M* and index *k* goes “around” *M* + 1 times while going

“up and down” *M* + *k* + 1 times. When the branch point is on the boundary, only half of

this surface is seen. The order of a boundary branch point must be even if the surface is

to touch the boundary monotonically–with odd order, it would double back on itself. Thus order 2

is the simplest case. Index 0 would give a piece of a plane, so index 1 is the simplest three-dimensional

case. Thus order 2, index 1 goes around (as an interior branch point) 3 times while going up and down 4 times;

and as a boundary branch point, it goes around one and a half times while going up and down twice. An

order 4, index 2 boundary branch point goes around two and a half times while going up and down three and

a half times. These are the cases illustrated at the links above.

The surfaces illustrated have simple formulas using a complex variable *z*. Namely, if

the surface is given by three functions X(z), Y(z), and Z(z),

then we have

X+iY = z^{M + 1}/(M+1)

Z = Re[ z^{M + k + 1}/(M+ k1)]

Here Re means “the real part of”. The first formula, which does not mention the real part explicitly,

is just a way of saying that X and Y are the real and imaginary parts, respectively, of a constant

times z^{m}, so they are (up to a constant) r^{M+1} cos((M+1)θ) and

r^{M+1} sin((M+1) θ), where r and θ are the

polar coordinates of z. These formulas corroborate the English descriptions above using the

phrase “goes around” so many times. The pictures above show the part of the surface defined over the disk |z|

≤ 0.6.

Here is a good problem for the beginner at this point: How many lines of intersection of the surface with

itself will there be, in a branch point of order M and index k?

Any branch point must be given by formulas that start out like those above, but there may also be terms with

higher powers of z. If *M+1* exactly divides *M+k+1* then to the first approximation, the surface will trace over

itself exactly. (You will have noticed that if you tried to solve the exercise above.)

If there is to be any separation of the different “sheets” of the surface, that must

arise from higher-order terms in the formula for the surface. You can see that in the following

picture, which shows a boundary branch point of order 2 and index 3, so M + 1 is 3 and

M + k + 1 is 6. Notice that at first glance the surface appears to be like a

distorted disk–but looking carefully, you can see from the coordinate markings on the surface

that it actually does go around one and half times, with the last half time exactly overlapping

the first half.

Order 2, Index 3 A false branch point.

If the overlap is exact, as in this picture, the branch point is called a false branch point. But bear in mind that the picture could look almost

like this, with the sheets being separated by a very tiny amount (that

might not even be visible in a computer-graphic picture) due to a term

in a high power of z. A branch point that is not false is called, naturally, a true branch point.

Here’s one last, more complicated, picture:

There exists locally isometric transformation between helicoid and catenoid.

This is because these surfaces have same Gaussian curvature at each points.

**Multi-resolution method for 3D multi-material topology optimization**

3D topology (In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.) optimization problems are solved with multi-material consideration. Incorporating multi-resolution scheme, this new method generates higher resolution designs in an efficient manner. Multi-material consideration bring limited complexity but also opens up new opportunities for designers and engineers as it suggests potential design alternatives that are not intuitive. **[]**

- J. Park,
**A. Sutradhar**, ‘A Multi-resolution Method for 3D Multi-material Topology Optimization’,*Computer Methods in Applied Mechanics and Engineering**,*Volume 285, 2015, Pages 571–586*.*(**Top 25 Hottest articles, Science direct**) (Impact factor 2.959) - J. Park,
**A. Sutradhar**, Design of 3D microstructures of isotropic materials for targeted elastic property using topology optimization,*Advances in Mechanical Engineering**,*2015*(in review).* - J. Park,
**A.****Sutradhar**, Efficient heat dissipation structures using multi-material and multi-resolution topology optimization,*International Journal of**Solids and Structures*, 2015*(in review)*