Saturday, April 30, 2011

War of the Octonions

Forgot the String Wars, sometimes called the Loop vs String Wars, or better known in the modern era as the Woit-Motl War? Well, fret not, it's back! At the center of the action is a piece written by John Baez and John Huerta in the current issue of Scientific American.

Of course, Peter Woit and Lubos Motl weigh in. Why not?

Peter Woit's take, and some very interesting responses by John Baez, can be found at Woit's weblog Not Even Wrong, here .

Lubos Motl's strong disagreements at his weblog The Reference Frame can be found, here.

For a brief review (or an elementary education, even) for those who don't know what an Octonion is, it's one of 4 division algebras, the first one being the ordinary elementary algebra taught in high school (REAL), the second (COMPLEX) known to Calculus students and Engineers and Scientists worldwide, the other two (Quaternions and Octonions) being a bit more involved. For comparison purposes, they take the forms:

R = a

C = a + bi

H = a + bi + cj + dk    

O = a + e1i  + e2j +e3k + e4l + e5m + e6n + e7o  , or something.

In any event, it's just notation folks, don't let it scare you.

The following probably has nothing to do with the above, but I ran into it looking for a cool picture for this post, rather than say just copying Lubos'. It's from Marni Dee Sheppeard's Arcadian Pseudofunctor: Feb. 7, 2011:

Theory Update 61

In this supercool paper, the authors define a tripled Fano plane(yes, that's right, three copies of Furey's particle zoo). It describes a set of 21=3×7 (left cyclic) modules over a noncommutative ring on eight elements. The ring is given by the upper triangular 2×2 matrices over the field with two elements. Similarly for right cyclic modules. 
The authors are familiar with the connection between octonion physics and so called stringy black holes. They find it odd that this structure is not studied in physics.

Just One Ticket Left for the 2015 Tourist Flyby of the Moon

Just One ($150 Million) Seat Remains on Space Adventures' Lunar Flyby

One of the two available tickets on Space Adventures' planned 2015 flyby of the moon has been sold, astronaut Byron Lichtenberg confirmed at an MIT conference today. If you're sitting on a small fortune and want to see the far side of the moon, act fast before the last seat on the Soyuz spacecraft is gone.

Read more: Space Adventures Lunar Flyby - The Future of Exploration MIT Conference - Popular Mechanics 

If you've got $150 million to spare and want to take a trip around the moon, don't wait for much longer—just one of the two seats that private space firm Space Adventures is selling for a proposed lunar flyby remains.

Byron Lichtenberg, a payload specialist aboard the space shuttle missions STS-9 and STS-45, noted this today at the MIT conference "Earth, Air, Ocean and Space: The Future of Exploration." The news that Space Adventures had sold one of the two nine-figure tickets came out quietly in January, when company CEO Eric Anderson mentioned it at a conference in Munich, Germany. Today, Lichtenberg, who had been in contact with the Space Adventures team, told a roomful of space pros, including astronauts Buzz Aldrin and Michael Massimino, that the sale was definite, sending a murmur of excitement through the room. Lichtenberg confirmed the statement with PM Senior Editor Joe Pappalardo, who is reporting from the conference.

Space Adventures, a Virginia-based company, has been planning a lunar flyby since 2005. It offered the two seats aboard a Russian-made Soyuz spacecraft that will fly around the moon in a mission scheduled for 2015. Anderson won't say who purchased the first $150 million ticket, but hinted that you'll know the person's name when you hear it.

What could a potential space traveler expect if they purchased the last remaining seat on Space Adventures' moon flyby? The no-frills Soyuz TMA carries one pilot and two passengers. It launches on a three-stage rocket, and will require extra propulsion for a moon flyby. After the Soyuz is launched, a second launch will send a rocket booster into low Earth orbit to rendezvous with the Soyuz and provide the addition propellant. It's the extra fuel and equipment needed to travel a quarter of a million miles—as opposed to simply journeying to the International Space Station or into orbit—that causes the insanely high price of the lunar trip.

Don't have $150 million? PM's May cover story, "The Early Adopter's Guide to Space Travel," shows you all the ways you'll be able to visit space in the not-too-distant future, including a few that will go a little easier on your wallet. And, PM predicts, even the lunar flyby will become a bit more affordable in the years to come—technological breakthroughs will bring down the trip's cost into the low millions. Just wait a few years.

Read more: Space Adventures Lunar Flyby - The Future of Exploration MIT Conference - Popular Mechanics

Friday, April 29, 2011

Hausdorff dimension

Estimating the Hausdorff dimension of the coast of Great Britain

In mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negativereal number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional vector space equals n. This means, for example the Hausdorff dimension of a point is zero, the Hausdorff dimension of a line is one, and the Hausdorff dimension of the plane is two. There are however many irregular sets that have noninteger Hausdorff dimension. The concept was introduced in 1918 by the mathematicianFelix Hausdorff. Many of the technical developments used to compute the Hausdorff dimension for highly irregular sets were obtained by Abram Samoilovitch Besicovitch.

Sierpinski triangle. A space with fractal dimension log 3 / log 2, which is approximately 1.585


[edit]Informal discussion

Intuitively, the dimension of a set (for example, a subset of Euclidean space) is the number of independent parameters needed to describe a point in the set. One mathematical concept which closely models this idea is that of topological dimension of a set. For example a point in the plane is described by two independent parameters (the Cartesian coordinates of the point), so in this sense, the plane is two-dimensional. As one would expect, the topological dimension is always a natural number.
However, topological dimension behaves in quite unexpected ways on certain highly irregular sets such as fractals. For example, the Cantor set has topological dimension zero, but in some sense it behaves as a higher dimensional space. Hausdorff dimension gives another way to define dimension, which takes the metric into account.
To define the Hausdorff dimension for a metric space X as a non-negative real number (that is, a number in the half-closed infinite interval [0, ∞)), we first consider the number N(r) of balls of radius at most r required to cover X completely. Clearly, as r gets smaller N(r) gets larger. Very roughly, if N(r) grows in the same way as 1/rd as r is squeezed down towards zero, then we say X has dimension d. In fact the rigorous definition of Hausdorff dimension is somewhat roundabout, as it allows the covering of X by balls of different sizes.
For many shapes that are often considered in mathematics, physics and other disciplines, the Hausdorff dimension is an integer. However, sets with non-integer Hausdorff dimension are important and prevalent.Benoît Mandelbrot, a popularizer of fractals, advocated that most shapes found in nature are fractals with non-integer dimension, explaining that "clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." [1]
There are various closely related notions of possibly fractional dimension. For example box-counting dimension, generalizes the idea of counting the squares of graph paper in which a point of X can be found, as the size of the squares is made smaller and smaller. (The box-counting dimension is also called the Minkowski-Bouligand dimension). The packing dimension is yet another notion of dimension admitting fractional values. These notions (packing dimension, Hausdorff dimension, Minkowski-Bouligand dimension) all give the same value for many shapes, but there are well documented exceptions.

]Formal definition

Let X be a metric space. If S\subset X and d\in[0,\infty), the d-dimensional Hausdorff content of S is defined by
C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i>0\Bigr\}.
In other words, C_H^d(S) is the infimum of the set of numbers \delta\ge 0 such that there is some (indexed) collection of balls \{B(x_i,r_i):i\in I\} covering S with ri > 0 for each i\in I which satisfies \sum_{i\in I}r_i^d<\delta. (One can assume, with no loss of generality, that the index set I is the natural numbers \mathbb N. Here, we use the standard convention that inf Ø =∞.) The Hausdorff dimension of X is defined by
\operatorname{dim}_{\operatorname{H}}(X):=\inf\{d\ge 0: C_H^d(X)=0\}.
Equivalently, \operatorname{dim}_{\operatorname{H}}(X) may be defined as the infimum of the set of d\in[0,\infty) such that the d-dimensional Hausdorff measure of X is zero. This is the same as the supremum of the set of d\in[0,\infty)such that the d-dimensional Hausdorff measure of X is infinite (except that when this latter set of numbers d is empty the Hausdorff dimension is zero).


  • The Euclidean space Rn has Hausdorff dimension n.
  • The circle S1 has Hausdorff dimension 1.
  • Countable sets have Hausdorff dimension 0.
  • Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension. For example, the Cantor set (a zero-dimensional topological space) is a union of two copies of itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff dimension is ln2 / ln3, which is approximately 0.63 The Sierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln3 / ln2, which is approximately 1.58.
  • Space-filling curves like the Peano and the Sierpiński curve have the same Hausdorff dimension as the space they fill.
  • The trajectory of Brownian motion in dimension 2 and above has Hausdorff dimension 2 almost surely.
  • An early paper by Benoit Mandelbrot entitled How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension and subsequent work by other authors have claimed that the Hausdorff dimension of many coastlines can be estimated. Their results have varied from 1.02 for the coastline of South Africa to 1.25 for the west coast of Great Britain. However, 'fractal dimensions' of coastlines and many other natural phenomena are largely heuristic and cannot be regarded rigorously as a Hausdorff dimension. It is based on scaling properties of coastlines at a large range of scales, but which does not however include all arbitrarily small scales, where measurements would depend on atomic and sub-atomic structures, and are not well defined.
  • The bond system of an amorphous solid changes its Hausdorff dimension from Euclidian 3 below glass transition temperature Tg (where the amorphous material is solid), to fractal 2.55±0.05 above Tg, where the amorphous material is liquid.[2]

[]Properties of Hausdorff dimension

[]Hausdorff dimension and inductive dimension

Let X be an arbitrary separable metric space. There is a topological notion of inductive dimension for X which is defined recursively. It is always an integer (or +∞) and is denoted dimind(X).
Theorem. Suppose X is non-empty. Then
 \operatorname{dim}_{\mathrm{Haus}}(X) \geq \operatorname{dim}_{\mathrm{ind}}(X).
 \inf_Y \operatorname{dim}_{\mathrm{Haus}}(Y) =\operatorname{dim}_{\mathrm{ind}}(X)
where Y ranges over metric spaces homeomorphic to X. In other words, X and Y have the same underlying set of points and the metric dY of Y is topologically equivalent to dX.
These results were originally established by Edward Szpilrajn (1907–1976). The treatment in Chapter VII of the Hurewicz and Wallman reference is particularly recommended.

[]Hausdorff dimension and Minkowski dimension

The Minkowski dimension is similar to the Hausdorff dimension, except that it is not associated with a measure. The Minkowski dimension of a set is at least as large as the Hausdorff dimension. In many situations, they are equal. However, the set of rational points in [0,1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.

[]Hausdorff dimensions and Frostman measures

If there is a measure μ defined on Borel subsets of a metric space X such that μ(X) > 0 and \mu(B(x,r))\le r^s holds for some constant s > 0 and for every ball B(x,r) in X, then  \operatorname{dim}_{\mathrm{Haus}}(X) \geq s. A partial converse is provided by Frostman's lemmaThat article also discusses another useful characterization of the Hausdorff dimension.

[]Behaviour under unions and products

If X=\bigcup_{i\in I}X_i is a finite or countable union, then
 \operatorname{dim}_{\mathrm{Haus}}(X) =\sup_{i\in I}  \operatorname{dim}_{\mathrm{Haus}}(X_i).
This can be verified directly from the definition.
If X and Y are metric spaces, then the Hausdorff dimension of their product satisfies[3]
 \operatorname{dim}_{\mathrm{Haus}}(X\times Y)\ge \operatorname{dim}_{\mathrm{Haus}}(X)+ \operatorname{dim}_{\mathrm{Haus}}(Y).
This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1.[4] In the opposite direction, it is known that when X and Y are Borel subsets of \R^n, the Hausdorff dimension of X\times Y is bounded from above by the Hausdorff dimension of X plus the upper packing dimension of Y. These facts are discussed in Mattila (1995).

[]Self-similar sets

Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set E is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(E) = E, although the exact definition is given below.
Theorem. Suppose
 \psi_i: \mathbb{R}^n \rightarrow \mathbb{R}^n, \quad i=1, \ldots , m
are contractive mappings on Rn with contraction constant rj < 1. Then there is a unique non-empty compact set A such that
 A = \bigcup_{i=1}^m \psi_i (A).
The theorem follows from Stefan Banach's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of Rn with the Hausdorff distance.[5]
To determine the dimension of the self-similar set A (in certain cases), we need a technical condition called the open set condition on the sequence of contractions ψi which is stated as follows: There is a relatively compact open set V such that
 \bigcup_{i=1}^m\psi_i (V) \subseteq V
where the sets in union on the left are pairwise disjoint.
Theorem. Suppose the open set condition holds and each ψi is a similitude, that is a composition of an isometry and a dilation around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is s where s is the unique solution of
 \sum_{i=1}^m r_i^s = 1.
Note that the contraction coefficient of a similitude is the magnitude of the dilation.
We can use this theorem to compute the Hausdorff dimension of the Sierpinski triangle (or sometimes called Sierpinski gasket). Consider three non-collinear points a1a2a3 in the plane R² and let ψi be the dilation of ratio 1/2 around ai. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket and the dimension s is the unique solution of
 \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1.
Taking natural logarithms of both sides of the above equation, we can solve for s, that is:
 s = \frac{\ln 3}{\ln 2}.
The Sierpinski gasket is self-similar. In general a set E which is a fixed point of a mapping
 A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A)
is self-similar if and only if the intersections
 H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0
where s is the Hausdorff dimension of E and Hs denotes Hausdorff measure. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally:
Theorem. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.

[]See also

[]Historical references


  1. ^ Mandelbrot, Benoît (1982). The Fractal Geometry of Nature. Lecture notes in mathematics 1358. W. H. Freeman. ISBN 0716711869.
  2. ^ M.I. Ojovan, W.E. Lee. (2006). "Topologically disordered systems at the glass transition"J. Phys.: Condensed Matter 18: 11507–20. doi:10.1088/0953-8984/18/50/007.
  3. ^ Marstrand, J. M. (1954). "The dimension of Cartesian product sets". Proc. Cambridge Philos. Soc. 50 (3): 198–202. doi:10.1017/S0305004100029236.
  4. ^ Falconer, Kenneth J. (2003). Fractal geometry. Mathematical foundations and applications. John Wiley & Sons, Inc., Hoboken, New Jersey.
  5. ^ Falconer, K. J. (1985). "Theorem 8.3". The Geometry of Fractal Sets. Cambridge, UK: Cambridge University Press. ISBN 0-521-25694-1.