Calabi-Yau manifold

If it is seen in a three dimensional space, the Calabi-Yau manifold will look like this picture.

A Calabi-Yau space is a mathematical construction used by physicists to describe parts of nature that are too small to see with the human eye.

Most people know that there are three space directions and one time direction in the universe – these directions are called dimensions. Physicists use Calabi-Yau spaces in studying high energy physics of which string theory is a part, to add 6 or 7 or other numbers to build up more dimensions to the universe.

The study of Calabi-Yau spaces is part of a mathematical theory known as "manifold theory".

What we know by observation of very small distances where Calabi-Yau takes place is pieced together by physicists.

We believe that matter is composed of atoms, and the distances of the pieces of the atoms exist at 10⁻¹⁵ meters. That is 0.000000000000001 meters. But wait. These tiny parts are also interesting. Why? Although they are small, they are also very high in energy. It is like they are compressed like a spring, or like a very long rubber band wrapped around a golf ball.

Now it might seem logical to think that the smaller we go, the less tight – that is, less energy. But physicists find that the smaller we go the more energy the tiny particles contain. They believe that is the size where these small but high energies begin to wrap themselves together much like the rubber band around the golf ball we talked about.

An experiment is going on in Geneva, Switzerland, by a group called CERN (http://public.web.cern.ch/Public/Welcome.html ). One of CERN's experiments will move groups of these particles extremely fast and let them smash together. Then they will watch the pieces of them as they fly apart.

When we collect pictures of the smash, we study them to see what happened. This process takes place in the world's largest "atom smasher", or as it is referred to today, a collider, and this one is called the LHC.

The experiments are exciting because physicists hope to find something never produced in experiments before called the Higgs particle ( http://www.exploratorium.edu/origins/cern/ideas/higgs.html), and some people think that little black holes may be made inside the LHC (http://public.web.cern.ch/PUBLIC/en/LHC/Safety-en.html).

But it is also believed we stand a chance at finding pieces of atoms and pieces called particles unwrapping into Calabi-Yau manifolds – or perhaps they will take other forms. Imagine if the golf ball were unwound really fast. The long rubber band would wobble and spin as it unraveled. This is exactly what it looks like when we look at the atomic particles in our collectors: http://cdsweb.cern.ch/record/39451.

Now this distance of 0.000000000000001 meters or 1/100,000,000,000,000 of a meter – is in wide consideration in physics. For example, the legendary physicist John Wheeler (1911–2008) can be seen working on maths with this distance where Calabi-Yau takes place here.(http://asymptotia.com/2008/04/14/john-wheeler-1911-2008/ ).

For the scales are tremendous and could be said to almost defy human perception. How small this is can be presented in this example. If we take a person to be on the average 1 and a half to 2 and a half meters tall – we consider the Calabi-Yau distances to be about 2 million million million times smaller than a human.

And for the hyper example where the universe as we know it completely breaks up into pure quantum ( 10^−33 centimetres or 10 ^ −35 metres in scientific notation),we need to go another 18 orders of magnitude smaller! What is this number? It is another TWENTY decimal places even smaller than the Calabi-Yau distance thresholds:

Calabi-Yau is theorized as beginning to occur at perhaps 1/1000000000000000 of a meter, as discussed above.

and

Quantum takes place completely at

1/100000000000000000000 meters distance BEYOND the atomic and Calabi-Yau scales of

1/1000000000000000 meters.

That IS small.

Now let us consider what space may look like at these Calabi-Yau distances. The model we learn at school for the most part is that it is all straight and 3 + 1 dimensional. ( http://www.javaview.de/molecule/figures/dna_sqr200.jpg )What we have studied about molecules for example fits 3 + 1 extremely well. (http://www.javaview.de/ ) Yet the data we often get in high energy experiments in colliders seem to suggest that under the presence of high energies space and time are bent around each other ( http://www.bathsheba.com/math/schwartzd/ ).

Now in order to get a better handle on manifolds, of which Calabi-Yau are an important set to high energy physics, let us begin with a Moebius strip. Here is a very practical presentation of a Moebius strip which is fun to do in classes introducing maths and physics: http://mathforum.org/sum95/math_and/moebius/moebius.html .

A Moebius strip has a half a dimension. Consider the paradox in our 3 space dimensions with one time dimension. Space as we recognize it here in our everyday life has three dimensions, time has one dimension, but there are no half dimensions.

But what happens if Moebius strip is cut right down the middle with a pair of scissors? The strip will form into one piece. Here in the world of half dimensions or boundaries, we find that when we cut something apparently in half, we come up with one.

This set of paradoxical behaviour was not unobserved by the great Dutch artist of the 20th century Mauritz Cornelius Escher: http://www.worldofescher.com/store/jpgs/P17L.jpg who portrayed in mastery and detail the shape of surfaces with the dimensions – or boundaries – cut, missing, overlapping and so forth in the most interesting unexpected and entertaining ways.

Now when we consider great energies and small distances it is not difficult to imagine how we find that things really curve up onto each other. Escher himself imagined this perhaps in the picture: http://www.facade.com/celebrity/photo/M_C_Escher.jpg .

Do we find curvatures such as this in nature? Absolutely. For example, we notice that a magnetic field looks like so many circles when we place iron shavings on a piece of paper around a magnet: http://www.gutenberg.org/files/16593/16593-h/images/image_224.jpg . And as if that were not enough we notice the same effect with respect to electricity: http://www.gutenberg.org/files/16593/16593-h/images/image_227.jpg .

And we also note that planets and planetary objects such as the Moon rotate about each other. Of course this is called the effect or force of gravity.

So we could say that the distance of 10^−15 meters, that is, 0.000000000000001 meters, is extremely important because it is the distance where the forces of nature in our classical or everyday world become less effective. This is the Calabi-Yau area or distance. At the same time more of the world at the small level, perhaps the level of say, dust, becomes much more important. The number of dimensions at Calabi-Yau is a subject of great interest to high energy physics. It could be called the nexus of high energy physics and classical physics. For example, at the cutting edge of the work where complex mathematics such as branes and supersymmetry emerge, one can read the papers of theoretical physicists such as Ed Witten and others at this excellent reference site: http://insti.physics.sunysb.edu/ITP/conf/simonswork3/talks.html.