Geometric cell division

Take a square and add a similar square to one of its sides and remove the shared edge. What remains is a rectangle.

Take an octagon and add a similar octagon to one of its sides and remove the shared edge. What remains is the following polygon.

 Take an n-gon and add a similar n-gon to one of its sides and remove the shared edge. At infinity the n-gons will divide into two circles. Unfortunately my computer is too slow to effectively record this in a video.

n=12

n=32

n=64

Regular polygons, exercise.

With a drawing program yesterday's pictures are easy to fake, of course. But these drawing programs don't give you the numbers.

Exercise.

Place a decagon edge-to-edge on a square with sides of length 1 ( see figure ). What is the distance between the two marked points?

( Answer: $ \frac{1}{4} \left(3 \sqrt{10-2 \sqrt{5}}+\sqrt{50-10 \sqrt{5}}+4\right) $ )

The mathematics of beauty.

A mathematics of beauty and art. There is no such thing one would expect. Yet, Owen Jones (1809 - 1874), London architect, wrote the 'General Principles in the Arrangement of Form and Colour, in Architecture and the Decorative Arts ... ' in his book The Grammar of Ornament.

Proposition 4.
True beauty results from that repose which the mind feels when the eye, the intellect and the affections are satisfied from the absence of any want.

from Page PL. I

You'll find the other propositions at the link above in the Digital Copy of the 'The Grammar of Ornament'.

From Intelligent Design to Geometry

Just some thoughts...

When I express my doubts about the success of the Apollo project people smile behind my back but that's about it. Friendships aren't broken and neither are ( employment ) contracts. ( See: AULIS on Apollofor more info ). In the last century science turned into a multi billion dollar industry and that certainly has changed the world of academics. Questioning a general accepted theory can ( and will ) ruin careers. Examples are questioning the cause of Aids, and questioning Darwin's Evolution Theory.

Evolution Theory basically says that we evolved over time by a process called mutation and natural selection. Those who question the Evolution Theory in fact question that very, very complex machines evolved from 'mud'. Darwin didn't answer that question because when Darwin published 'On the Origin of Species on 24 November 1859' he wasn't even remotely aware of the complexity of the cells making up life. DNA wasn't discovered until ten years later in 1869 by Friederich Miescher and it took until 1953 when James Watson and Francis Crick discovered the double helix structure of DNA. Intelligent Design Theory accept Evolution Theory but only up to a certain point. They argue that somewhere in the beginning some information or 'design' had to be injected into the system. Who put it there? I would ask.

To the point.

Anyway, these thoughts entered my mind because I am thinking of building a 'DNA-type-of' geometry building block for a computer program. With the help of a computer these geometries should be able to construct ( divide ) themselves in a scene graph and evolve, multiply and so on. At the moment it's just an idea. I started to look for a way to understand more about DNA by finding popular science books on the subject. I haven't learned much about biology and chemistry and what I have learned seems forgotten. But I am only interested in DNA as a computer, or data structure. Then I found this website 'DNA seen through the eyes of a coder'. Since I am a coder ( computer, Android, programmer ) by profession that was exactly what I was looking for. Take this for example, DNA is not binary, DNA is quaternary. Computer letters normally consist of 8 bits called a byte, so using that system there are 256 possible letters. The equivalent of a byte in DNA is the codon and has three places. So in DNA language there are 64 possible letters. Read more on the site.

More to follow on this geometry project soon, I expect.

jenn3d

I came across a free tool for visualizing Coxeter polytopes, jenn3d. I suppose this program visualizes the Coxeter groups of polytopes. Polytopes are geometric objects in the n-th dimension with flat sides.



Link: jenn3d.org

Educational tools for symmetry

#graphica# #m336#

This website from Prof. Dr. Loren Williams is worth exploring. Don't forget to visit the Crystallography Tool section.

P.S.
Notice the man in the space suit

Unexpected behavior of C2mm symmetry.

#openuniversity# #m336# #symmetry#

I read that a US Senate Commission is worried about the F-35 project because they are building planes while the testing of the F-35 is in full progress. The thing is that the Joint Strike Fighter ( F-35 ) is in fact a flying super-computer running on state-of-the-art software. Sound engineering principles don't apply to these machines. Software is never 'done'. Software is always in development and in testing and ( just ) released at the same time. There will be upgrades for the F-35 until the end of its life. Politicians think ( or say they think ) that when the plane is done, its done.

This thought crossed my mind because planning, by definition, implies uncertainty about the future. Unexpected things can happen, will happen, at a moment when its least expected.

To the point.

Mathematics is unpredictable too. From time to time you'll see unexpected things. Among various other topics I am studying plane symmetries at the moment. I have several books well illustrated with all sorts of patterns that can occur. For me, programming is an effective way to study, so I wrote a program that plots patterns using the symmetries I am studying. One of these symmetries is C2mm which is basically rotating a diamond lattice 90, 180, 270 and 360 degrees. While I was testing the C2mm symmetry in Graphica ( the name of my symmetry program ) I noticed that the patterns are very sensitive to the center of rotation.

I made a video ( of only part of the screen for size and performance reasons ). The second half of the video shows several unexpected patterns while changing the center of rotation. Watch and you may experience the same awe that I felt. All I expected was that the symmetry could generate a diamond lattice from a triangle.

Open University M336 video lectures

#M336 #geometry #openuniversity

The Open University M336 course comes with 7 lectures on one DVD of about half an hour each, or almost four hours of lectures. The lectures are titled:
- Living with patterns
- Friezes
- Counting with groups
- Incidence symbols
- Lattices and wallpaper patterns
- Regular solids
- Octet for truss and comb

These lectures are additions to the booklets and excercises and are not meant to learn new material from, instead they reinforce what has been learned before.

So, although there is no entire lecture series covering the M336 materials you could easily create one by cherry picking lectures from the internet. For Group Theory you can use the first half of the Harvard Abstract Algebra course which covers group theory upto the Sylow Theorems.

For the geometry part you could use the MIT Course 'An Introduction to Crystallography'. This course contains 41 video lectures of which lectures 5 to 27 cover the material of the Geometry track in M336.

Link: Symmetry, Structure, and Tensor Properties of Materials MIT OpenCourseWare

Escher's imaginery workplace

#mathematics #art #Open University #m336 #Escher

The Scream by Edvard Munch was sold for USD 120 million. I didn't like it yesterday and I don't like it now that I know it's perceived value. My favourite artists are Escher, Kandinsky and Dali, their work inspires me, and I am truly impressed by what they have created, art needs beauty. M336 brings group theory and geometry together through visual symmetry, or the symmetry Escher used in a lot of his work. I browse a lot through work of Escher as a result of M336 studies. Recently I came across this sensational video. A must see, really.


( A short movie inspired on Escher's works and a free vision on how it could be his workplace. )

This is another video by Eterea.

( A short movie about numbers and geometry. )

M336 Groups and Geometry

Thirty point Open University courses consist of four blocks, where the sixty pointers have eight. M336 Groups and Geometry is a four block course. Since there are two inter-related but independent tracks it doesn't feel like an ordinary 30 point course, somewhat heavier in fact. The geometry course roughly discusses one topic per block:
Block 1: Frieze groups
Block 2: Tilings
Block 3: 2D-Lattices and wallpaper groups
Block 4: 3D-Lattices.

See this previous post about block1 and frieze groups.

I am almost done with block 2 but I am still struggling with tilings ( TMA02 question 4 ). In the meantime I have coded a nice Mathematica pattern editor, ( which I hope will form the base for a Wallpaper Group editor and generator ).

Click to enlarge.

Programming Mathematica is easy and fast, that is: after you have wrestled yourself through the rather steep learning curve. An advanced topic in Mathematica ( i.e. chapter 15 in the Cookbook ) is the programming with DynamicModules and Manipulate. It turns out that, even as a GUI, Mathematica seems to have no limitations to what is possible. In order to code the pattern editor I had to crash myself through Manipulate for which I received invaluable help from the Mathematica experts community at Mathematica StackExchange. Thank you very much!

What is Sacred Geometry ?

One of my ( many ) reasons to study mathematics is my fascination for the geometry of crop circles. Appreciating the art in the crop circles does not mean that I have an opinion on how, or by who, the circles are made. I found out that the more you get into the subject the harder it is to answer that question. Only people that know very little about them will say without hesitation that all circles are 'hoaxes'. Hoaxes or not, they are absolutely beautiful.

Bert Janssen, a Dutch crop researcher, looked into the geometry of some circles.

(c) Bert Janssen. Click to enlarge.

See also: crop circle geometry, by Bert Janssen

Another crop circle researcher is Lucy Pringle.

The Godfather of crop circle research is Colin Andrews, he started the crop circle community.

To the point: sacred geometry.

It is rather common to use that word sacred geometry in the crop circle community. But it is also used in the so called ancient aliens theory which explains the sudden appearance of high tech architecture like the Pyramids in Giza by visitations of extra-terrestrials in that period.

The word sacred means:
- devoted or dedicated to a deity or to some religious purpose;
- pertaining to or connected with religion.
The word geometry means:
- a branch of mathematics concerned with questions of shape, size, position of figures, etc.

In mathematics the field of geometry has many branches, i.e. Eucledian, non-Euclidean, algebraic, analytical and so on, but there is no mathematical branch of geometry called sacred geometry. My conclusion is therefore that:

Sacred geometry is the geometry that has been used in the construction of religious artifacts.

By coincidence or not, it turns out that these geometries are often related to the golden ratio which is considered to have special aesthetic qualities.

What is a lattice? - Or lattices in M336

#openuniversity #m336

The Open University course M336 contains two booklets which are dedicated to lattices. One booklet about two-dimensional lattices (GE3) and one about three-dimensional lattices ( and polyhedra ) (GE6). To a layman I would explain lattice as some regular grid of points ( connected by thin lines ).

Click to enlarge

In the example above the lattice is defined by two vectors and consists of all points $n \mathbf{a} + m \mathbf{b}$ where $n,m$ are integers.

Fields which use lattice theory are crystallography, finance, game ( maze ) programming, group theory and number theory. When I dug a little bit deeper I discovered that the field of lattices is -ginormous-. Gabriele Nebe and Neil Sloan ( yes him ) maintain a catalog of lattices which now contains over 160,000 lattices. Mathematicians like to generalize over n-dimensions so yes, that database contains lattices in dimensions higher than 3. Like lattices in 40 dimensions for example. Forty.

A catologue of lattices.
Junkyard article about lattices and geometry of numbers.

The mathematical universe is expanding with tremendous speed.

Frieze Patterns and Conway

#mathematica #m336 #openuniversity

John Horton Conway (26 December 1937 - ) is a prolific mathematician who contributed to many branches of mathematics. He is the inventor of the cellular automaton "Game of Life". He is currently Professor at Princeton University. He added yet another set of names to the Frieze Patterns. Since they are not mentioned in the M336 course booklet I suppose the names weren't adopted widely enough.

Conway proposed the following names for the seven frieze patterns:
- Hop for p111, translational ( only ).
- Sidle for pm11, vertical.
- Jump for p1m1, horizontal.
- Step for p1a1, glide.
- Spinning hop for p112 rotational.
- Spinning jump for pmm2 horizontal and vertical.
- Spinning sidle for pma2 vertical glide.

Click to enlarge

The squared wheel

#geometry

The squared wheel is a beautiful example of mathematical thinking: thinking out-of-the-box, generating alternatives and cutting one's way through the jungle. It is also a motivator to think deeply about a problem, even if it seems obvious.



If you have asked yourself the question: "Are there more wheel configurations possible than just the circle and the square?" then, I suppose, you got what it takes. ;-)

Have a look at the Wolfram demo 'Roads and Wheels'.

Drawing ( friezes ) with Bezier curves (2)

#OpenUniversity #M336

Of, course colors can be added and so forth which may generate some interesting problems for M336 GE1 Counting with Groups.

See also:

- 'Frieze group pmm2'
- Drawing ( friezes ) with Bezier curves

Drawing ( friezes ) with Bezier curves

#OpenUniversity #M336

Although the friezes in post 'Frieze group pmm2' consisted of only straight lines it is possible to create friezes containing Bezier curves in Mathematica using the BezierCurve function.

The following frieze will look familiar to M336 students. ;-)

Frieze group pmm2

#OpenUniversity little success story

The most common barrier to effective study is the so called 'Lack of Mass' ( Applied Scholastics ). A subject has not enough mass -for you- when you don't like it, aren't interested in it, can't see the purpose of studying it, etc.

If this situation occurs then you simply (...) have to 'add mass'. I did it for Open University course M336 IB3 Frieze Patterns by programming a frieze pattern tool in Mathematica. I like programming and if you can program a topic it is proof that you understand the topic. Now friezes live for me. I know them all, including the recognition algorithm.

Here are some applications of the tool I made.

A graphical proof that a frieze containing the letter H ( i.e. HHH... ) has symmetry group pmm2.

Or do it the other way around: take a letter R frieze and transform it to a frieze with p1a1 symmetry.

And now I can't wait to start with the Wallpaper Patterns. So, if you don't like a subject you can do two things: wait until you start liking it which may be never, or take creative action so that you -do- like it.

Symmetry ?

Browsing photo albums on facebook often reveals interesting and beautiful pictures.

(c) Yuliya Keaton

I find patterns like this very interesting because most people would call this a symmetric pattern. Yet, I know of no mathematical symmetry that accurately describes the symmetry we see here. Yet, mathematics claims to have categorized all symmetries of the plane.

Animated Penrose Tiling

I am strolling around in the world of 2D Euclidean geometry. What can be so interesting about something -that- "simple"? This is an area of mathematics which is deeply connected to human imagination and art. Think Escher. I came across a YouTube video I would like to share. Worth the watch.

An animation of the celebrated Penrose non periodic tiling made with Povray, realized at the Department of Mathematics and Physics, Catholic University, Brescia (Italy). By Maurizio Paolini and Alessandro Musesti.

Mandelbrot fractals in 3D

The laptops we use today are extremely powerful machines if you measure them against the standards of a decade ago. In these days producing ( rendering ) Mandelbrot fractals was hard work for any computer. Also, Mandelbrot fractals were 2D, by definition, case closed. Experimenting with '3d-type-of-Mandelbrots' was impossible due to the limitations of the hardware. A lot has happened since then. - Daniel White created a website about the topic, called 'The unravelling of the Real 3D Mandelbulb" where he explains the interesting ( and surprising ! ) history of 3D Mandelbrots.