## Monday, January 31, 2011

### Celebrating 53 Years of American Space Exploration

Happy 53rd birthday, American Space Program

Explorer 1 (1958 Alpha 1)[5] was the first Earth satellite of the United States, launched as part of its participation in the International Geophysical Year. The mission followed the first two Earth satellites the previous year, the Soviet Union's Sputnik 1 and 2, beginning the Cold War Space Race between the two nations.

Explorer 1 was launched on January 31, 1958 at 22:48 Eastern Time (this is equal to February 1, 03:48 UTC because the time change goes past midnight) atop the first Juno booster from LC-26 at the Cape Canaveral Missile Annex, Florida. It was the first spacecraft to detect the Van Allen radiation belt,[6] returning data until its batteries were exhausted after nearly four months. It remained in orbit until 1970, and has been followed by more than 90 scientific spacecraft in the Explorer series.

The U.S. Earth satellite program began in 1954 as a joint U.S. Army and U.S. Navy proposal, called Project Orbiter, to put a scientific satellite into orbit during the International Geophysical Year. The proposal, using a military Redstone missile, was rejected in 1955 by the Eisenhower administration in favor of the Navy's Project Vanguard, using a booster produced for civilian space launches.[7] Following the launch of the Soviet satellite Sputnik 1 on October 4, 1957, the initial Project Orbiter program was revived as the Explorer program to catch up with the Soviet Union.[8]

Explorer 1 was designed and built by the Jet Propulsion Laboratory (JPL), while a Jupiter-C rocket was modified by the Army Ballistic Missile Agency (ABMA) to accommodate a satellite payload; the resulting rocket known as the Juno I. The Jupiter-C design used for the launch had already been flight-tested in nose cone reentry tests for the Jupiter IRBM, and was modified into Juno I. Working closely together, ABMA and JPL completed the job of modifying the Jupiter-C and building Explorer 1 in 84 days. However, before work was completed, the Soviet Union launched a second satellite, Sputnik 2, on November 3, 1957. The U.S. Navy's attempt to put the first U.S. satellite into orbit failed with the launch of the Vanguard TV3 on December 6, 1957.[9]

 OOPS! The Vanguard TV3 rocket explodes 2 seconds and 4 ft. above the launch pad on Dec. 6, 1957. FAIL! Well, they can't all be winners. We got better. On March 17, 1958, Vanguard 1 became the second artificial satellite successfully placed in Earth orbit by the United States.
UPDATE: Thanks to the good folks at Universe Today weblog, we are reminded that today is also the 50th anniversary of American Chimp-o-naut Ham's successful suborbital flight, and the 40th anniversary of the launch of  Apollo 14

## Sunday, January 30, 2011

### 45 Days until Mercury Orbit Insertion

Humanity will chalk up another grand astronomical engineering achievement when the MESSENGER probe inserts itself into orbit around planet Mercury in 45 days.

 "Hello Mercury, it's me, MESSENGER." It looks like the moon, but it hides a stupid huge amount of vast exploitable riches

The date of insertion will be  March 18, 2011. For the first time we will be able to study in great detail, this, the least known planet*. MESSENGER will probe the surface for a year, but you know how these things go, so the mission will probably be running for years.

Click here for NASA's official page on Project MESSENGER

* - OK, OK if you're old school and still think Pluto is a planet, then technically we know less about Pluto and its moons Charon, Nix, and Hydra, but don't worry New Horizons is paying a visit and in 2015 we'll know more.

Mark my words: one hundred years from now there will be many fortunes made from the mining of Mercury. The Moon is so-o close and beckons first. Mars is for romantic dreamers and eventually we'll get there too. But Mercury is for businessmen who want to make a buck. Why?

In a word: heavy metals and mineral wealth. As the solar system formed, the heavier stuff moved in toward to Sun. The Sun itself has the greatest wealth in the heavy stuff and minerals, but that stuff is all at the Sun's core where we can't reach it for the next thousand years.

Mercury on the other hand is close at hand. The heavy stuff that was strongly drawn in toward the Sun's gravity well at a wide angle and didn't fall in coalesced into a very heavy planet very close by, indeed  ... the closest ...

Planet Mercury, richest of planets. A planet to make mining companies drool.

This achievement brought to you fellow humans by The United States of America, leading the way in astronomical engineering achievements since we figured out how to launched an Explorer satellite  on January 31, 1958, 53 years ago...

You're welcome. But no, thank you very much, for where would America be without you?

Here's something else from the USA that I think we can also all wrap at least our heads around. Pop the champagne, it's party time! Take it away, Joan! ....

### That's Small !

Click here for laughs ( the Cigar Guy)
 Human Hair (18 to 180 µm)

• 1.6 × 10−35 metres = the Planck length (lengths smaller than this do not make any physical sense, according to current theories of physics)
• ...........11 orders of magnitude larger

Click here for an orders of ten magnitude interactive visualization from the galactic to the quark.

In words
(long scale)
In words
(short scale)
Prefix Symbol Decimal Power
of ten
Order of
magnitude
quadrillionth septillionth yocto- y 0.000000000000000000000001 10−24 −24
trilliardth sextillionth zepto- z 0.000000000000000000001 10−21 −21
trillionth quintillionth atto- a 0.000000000000000001 10−18 −18
billiardth quadrillionth femto- f 0.000000000000001 10−15 −15
billionth trillionth pico- p 0.000000000001 10−12 −12
milliardth billionth nano- n 0.000000001 10−9 −9
millionth millionth micro- µ 0.000001 10−6 −6
thousandth thousandth milli- m 0.001 10−3 −3
hundredth hundredth centi- c 0.01 10−2 −2
tenth tenth deci- d 0.1 10−1 −1
one one - - 1 100 0
ten ten deca- da 10 101 1
hundred hundred hecto- h 100 102 2
thousand thousand kilo- k 1,000 103 3
million million mega- M 1,000,000 106 6
milliard billion giga- G 1,000,000,000 109 9
billion trillion tera- T 1,000,000,000,000 1012 12
billiard quadrillion peta- P 1,000,000,000,000,000 1015 15
trillion quintillion exa- E 1,000,000,000,000,000,000 1018 18
trilliard sextillion zetta- Z 1,000,000,000,000,000,000,000 1021 21
quadrillion septillion yotta- Y 1,000,000,000,000,000,000,000,000 1024 24
 King Tut's death mask is safe and sound. Looks good, too.

## Saturday, January 29, 2011

### Groups(2) - Mirror Symmetry and Rotational Symmetry

Evariste Galois died in a duel aged 20, but left behind enough ideas to keep mathematicians busy for centuries. These involved the theory of groups, mathematical constructs that can be used to quantify symmetry. Apart from its artistic appeal, symmetry is the essential ingredient for scientists who dream of a future theory of everything. Group theory is the glue which binds the 'everything' together."

... Tony Crilly

So begins the four page "Introduction" by Manchester UK Maths Historian Tony Crilly at the beginning of his four-page Chapter 38, Groups,  in his book "50 Mathematical Ideas You Really Need to  Know" We will return to the rest of his wonderful introduction momentarily, but first a few words about this weblog.

So far I have been posting in a stream-of-consciousness fashion, that is to say whatever topic interests me at any one time makes my weblog. However, as we dig deeper into "reality" we need to apply structure, so we will do so without getting carried away as so:

The subjects of my posts have recently begun to revolve and will continue to cycle through various fields that interest me, notably and in this order, for now:

- Mathematics
- Mathematical Physics/Applied Mathematics (same thing?)
- Physics
- Engineering
- Astronomy
- Lunar colonization

Today's lesson is on Group theory, which is vitally important in our real world which is best described by gauge theory, which grew from quantum field theory, which grew from quantum mechanics and special relativity. So let's begin.

Crilly continues:

Symmetry is all around us. Greek vases have it, snow crystals have it, buildings often have it and some letters of our alphabet have it. There are several sorts of symmetry: chief among them are mirror symmetry and rotational symmetry. We'll look at just two-dimensional symmetry for the purposes of this introduction - all our objects live on the flat surface of a two-dimensional plane.

MIRROR SYMMETRY

Can we set up a mirror so that an objects looks the same in front of the mirror as in the mirror? The word MUM has mirror symmetry, but HAM does not; MUM in front of the mirror is the same as MUM in the mirror while HAM  becomes MAH. A 2-D tripod has mirror symmetry, but the triskelion (tripod with feet) does not. The triskelion as the object before the mirror is right-handed but its mirror image in what is called the image plane is left-handed.

 Triskelion

ROTATIONAL SYMMETRY

We can also ask if there is an axis perpendicular to the page so the object can be rotated in the page through an angle and be brought back to its original position. Both the tripod and triskelion have rotational symmetry. The triskelion, meaning "three legs", is an interesting shape. The right-handed version is a figure which appears as the symbol of the Isle of Man  and also on the flag of Sicily.

If we rotate through 120 degrees and 240 degrees the rotated figure will coincide with itself; if you closed your eyes before rotating it you would see the same triskelion when you opened them again after rotation.

The curious thing about the three-legged figure is that no amount of rotation keeping in the plane will ever convert a right-handed triskelion into a left-handed one. Objects for which the image in the mirror is distinct from the object in front of the mirror are called chiral - they look similar but are not the same. The molecular structure of some chemical compounds may exist in some right-handed and left-handed forms in three dimensions and are examples of chiral objects. This is the case with the chemical compound limosene which in one form tastes like lemons and in the other case like oranges. The drug thalidomide in one form is an effective cure of morning sickness in pregnancy but in the other form has tragic consequences.

Steve here. That was only the first page and a half of Crilly's introduction. Next up in the rotation, probably next week at this time, will be Groups(3) - Measuring symmetry, being the continuation of Crilly. We will be introducing Cayley tables, which will give us the first opportunity to measure symmetry, based on the work of Arthur Cayley around 1854.

They're not hard, and will show the amazing result that the tripod, and not the triskelion, is the more complicated of the two.

If you're chomping at the bit and can't stand the suspense, or simply wish to read ahead, here's some voluntary non-required pre-homework for the elite of brain, being the entry on Cayley tables from the EVER-BORING and love-of-subject-killing textbooky encyclopedia that is mind-numbing Wikipedia:

A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a group — such as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's center — can be easily deduced by examining its Cayley table.
A simple example of a Cayley table is the one for the group {1, −1} under ordinary multiplication:

× 1 −1
1 1 −1
−1 −1 1

## History

Cayley tables were first presented in Cayley's 1854 paper, "On The Theory of Groups, as depending on the symbolic equation θ n = 1". In that paper they were referred to simply as tables, and were merely illustrative — they came to be known as Cayley tables later on, in honour of their creator.

## Structure and layout

Because many Cayley tables describe groups that are not abelian, the product ab with respect to the group's binary operation is not guaranteed to be equal to the product ba for all a and b in the group. In order to avoid confusion, the convention is that the first factor (termed nearer factor by Cayley) in any row of the table is the same, and that the second factor (or further factor) in any column is the same, as in the following example:

* a b c
a a2 ab ac
b ba b2 bc
c ca cb c2

Cayley originally set up his tables so that the identity element was first, obviating the need for the separate row and column headers featured in the example above. For example, they do not appear in the following table:

 a b c b c a c a b

In this example, the cyclic group Z3, a is the identity element, and thus appears in the top left corner of the table. It is easy to see, for example, that b2 = c and that cb = a. Despite this, most modern texts — and this article — include the row and column headers for added clarity.

## Properties and uses

### Commutativity

The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table is symmetric along its diagonal axis. The cyclic group of order 3, above, and {1, −1} under ordinary multiplication, also above, are both examples of abelian groups, and inspection of the symmetry of their Cayley tables verifies this. In contrast, the smallest non-abelian group, the dihedral group of order 6, does not have a symmetric Cayley table.

### Associativity

Because associativity is taken as an axiom when dealing with groups, it is often taken for granted when dealing with Cayley tables. However, Cayley tables can also be used to characterize the operation of a quasigroup, which does not assume associativity as an axiom (indeed, Cayley tables can be used to characterize the operation of any finite magma). Unfortunately, it is not generally possible to determine whether or not an operation is associative simply by glancing at its Cayley table, as is the case with commutativity. This is because associativity depends on a 3 term equation, (ab)c = a(bc), while the Cayley table shows 2-term products. However, Light's associativity test can determine associativity with less effort than brute force.

### Permutations

Because the cancellation property holds for groups (and indeed even quasigroups), no row or column of a Cayley table may contain the same element twice. Thus each row and column of the table is a permutation of all the elements in the group. This greatly restricts which Cayley tables could conceivably define a valid group operation.

To see why a row or column cannot contain the same element more than once, let a, x, and y all be elements of a group, with x and y distinct. Then in the row representing the element a, the column corresponding to x contains the product ax, and similarly the column corresponding to y contains the product ay. If these two products were equal — that is to say, row a contained the same element twice, our hypothesis — then ax would equal ay. But because the cancellation law holds, we can conclude that if ax = ay, then x = y, a contradiction. Therefore, our hypothesis is incorrect, and a row cannot contain the same element twice.

Exactly the same argument suffices to prove the column case, and so we conclude that each row and column contains no element more than once. Because the group is finite, the pigeonhole principle guarantees that each element of the group will be represented in each row and in each column exactly once.
Thus, the Cayley table of a group is an example of a latin square.

## Constructing Cayley tables

Because of the structure of groups, one can very often "fill in" Cayley tables that have missing elements, even without having a full characterization of the group operation in question. For example, because each row and column must contain every element in the group, if all elements are accounted for save one, and there is one blank spot, without knowing anything else about the group it is possible to conclude that the element unaccounted for must occupy the remaining blank space. It turns out that this and other observations about groups in general allow us to construct the Cayley tables of groups knowing very little about the group in question.

### The "identity skeleton" of a finite group

Because in any group, even a non-abelian group, every element commutes with its own inverse, it follows that the distribution of identity elements on the Cayley table will be symmetric across the table's diagonal. Those that lie on the diagonal are their own inverse; those that do not have another, unique inverse.

Because the order of the rows and columns of a Cayley table is in fact arbitrary, it is convenient to order them in the following manner: beginning with the group's identity element, which is always its own inverse, list first all the elements that are their own inverse, followed by pairs of inverses listed adjacent to each other.

Then, for a finite group of a particular order, it is easy to characterize its "identity skeleton", so named because the identity elements on the Cayley table are clustered about the main diagonal — either they lie directly on it, or they are one removed from it.

It is relatively trivial to prove that groups with different identity skeletons cannot be isomorphic, though the converse is not true (for instance, the cyclic group C8 and the quaternion group Q are non-isomorphic but have the same identity skeleton).

Consider a six-element group with elements e, a, b, c, d, and f. By convention, e is the group's identity element. Because the identity element is always its own inverse, and inverses are unique, the fact that there are 6 elements in this group means that at least one element other than e must be its own inverse. So we have the following possible skeletons:
• all elements are their own inverses,
• all elements save d and f are their own inverses, each of these latter two being the other's inverse,
• a is its own inverse, b and c are inverses, and d and f are inverses.
In our particular example, there does not exist a group of the first type of order 6; indeed, simply because a particular identity skeleton is conceivable does not in general mean that there exists a group that fits it.
It is noteworthy (and trivial to prove) that any group in which every element is its own inverse is abelian.

### Filling in the identity skeleton

Once a particular identity skeleton has been decided on, it is possible to begin filling out the Cayley table. For example, take the identity skeleton of a group of order 6 of the second type outlined above:

e a b c d f
e e

a
e

b

e

c

e

d

e
f

e

Obviously, the e row and the e column can be filled out immediately. Once this has been done, it may be necessary (and it is necessary, in our case) to make an assumption, which may later lead to a contradiction — meaning simply that our initial assumption was false. We will assume that ab = c. Then:

e a b c d f
e e a b c d f
a a e c

b b
e

c c

e

d d

e
f f

e

Multiplying ab = c on the left by a gives b = ac. Multiplying on the right by c gives bc = a. Multiplying ab = c on the right by b gives a = cb. Multiplying bc = a on the left by b gives c = ba, and multiplying that on the right by a gives ca = b. After filling these products into the table, we find that the ad and af are still unaccounted for in the a row; as we know that each element of the group must appear in each row exactly once, and that only d and f are unaccounted for, we know that ad must equal d or f; but it cannot equal d, because if it did, that would imply that a equaled e, when we know them to be distinct. Thus we have ad = f and af = d.

Furthermore, since the inverse of d is f, multiplying ad = f on the right by f gives a = f2. Multiplying this on the left by d gives us da = f. Multiplying this on the right by a, we have d = fa.
Filling in all of these products, the Cayley table now looks like this:

e a b c d f
e e a b c d f
a a e c b f d
b b c e a

c c b a e

d d f

e
f f d

e a

Because each row must have every element of the group represented exactly once, it is easy to see that the two blank spots in the b row must be occupied by d or f. However, if one examines the columns containing these two blank spots — the d and f columns — one finds that d and f have already been filled in on both, which means that regardless of how d and f are placed in row b, they will always violate the permutation rule. Because our algebraic deductions up until this point were sound, we can only conclude that our earlier, baseless assumption that ab = c was, in fact, false. Essentially, we guessed and we guessed incorrectly. We, have, however, learned something: abc.

The only two remaining possibilities then are that ab = d or that ab = f; we would expect these two guesses to each have the same outcome, up to isomorphism, because d and f are inverses of each other and the letters that represent them are inherently arbitrary anyway. So without loss of generality, take ab = d. If we arrive at another contradiction, we must assume that no group of order 6 has the identity skeleton we started with, as we will have exhausted all possibilities.
Here is the new Cayley table:

e a b c d f
e e a b c d f
a a e d

b b
e

c c

e

d d

e
f f

e

Multiplying ab = d on the left by a, we have b = ad. Right multiplication by f gives bf = a, and left multiplication by b gives f = ba. Multiplying on the right by a we then have fa = b, and left multiplication by d then yields a = db. Filling in the Cayley table, we now have (new additions in red):

e a b c d f
e e a b c d f
a a e d
b
b b f e

a
c c

e

d d
a

e
f f b

e

Since the a row is missing c and f and since af cannot equal f (or a would be equal to e, when we know them to be distinct), we can conclude that af = c. Left multiplication by a then yields f = ac, which we may multiply on the right by c to give us fc = a. Multiplying this on the left by d gives us c = da, which we can multiply on the right by a to obtain ca = d. Similarly, multiplying af = c on the right by d gives us a = cd. Updating the table, we have the following, with the most recent changes in blue:

e a b c d f
e e a b c d f
a a e d f b c
b b f e

a
c c d
e a
d d c a

e
f f b
a e

Since the b row is missing c and d, and since b c cannot equal c, it follows that b c = d, and therefore b d must equal c. Multiplying on the right by f this gives us b = cf, which we can further manipulate into cb = f by multiplying by c on the left. By similar logic we can deduce that c = fb and that dc = b. Filling these in, we have (with the latest additions in green):

e a b c d f
e e a b c d f
a a e d f b c
b b f e d c a
c c d f e a b
d d c a b
e
f f b c a e

Since the d row is missing only f, we know d2 = f, and thus f2 = d. As we have managed to fill in the whole table without obtaining a contradiction, we have found a group of order 6: inspection reveals it to be non-abelian. This group is in fact the smallest non-abelian group, the dihedral group D3:

* e a b c d f
e e a b c d f
a a e d f b c
b b f e d c a
c c d f e a b
d d c a b f e
f f b c a e d

## Generalizations

The above properties depend on some axioms valid for groups. It is natural to consider Cayley tables for other algebraic structures, such as for semigroups, quasigroups, and magmas, but some of the properties above do not hold.

Groups(1) - Gross Introduction

## Friday, January 28, 2011

### The Amazing Tetrahedron

There are only 5 Platonic solids, and the simplest of the 5 is the 4-cornered (vertices), 4-faced, 4-sided shape known as the Tetrahedron.

So what's the big deal, right? It looks pretty simple, so it must be uninteresting, right?

I thought so, but for such a simple shape, it's actually much more interesting than it looks.

First, a cool explanation of it's name, from Pat Ballew's Math Words::

Tetrahedron A tetrahedron is the most simple of three space shapes since it consists of only four vertices (see figure below). The Greek tetra stands for four, and can still be found in some science words such as tetrachloride or tetravalent. The hedra is from the Greek for base, or seat.

OK, now that we know what it is, what can we do with it? What properties does it have that one may call: Interesting?

For starters, check out the WolframMathWorld article on Dual Polyhedra, that is to say, if you put a plane on a polygon where a point exists, what new shape do you get, that is to say, what is its dual?

As it turns out, the dual of a cube is an octagon and vice versa, and the dual of an icosahedron is a dodecahedron and vice versa, but the dual of a tetrahedron is ... another tetrahedron!

In short, a tetrahedron is its own dual, as illustrated below:

"The process of forming duals is illustrated above for the Platonic solids. The top row shows the original solids. The middle row shows the vertex figures of the original solid as lines superposed on the tangential polygons forming the corresponding duals. Finally, the polyhedron compounds consisting of a polyhedron and its dual are illustrated in the bottom row." ... WolframMathWorld

The shape in the lower left-hand corner, being the polyhedron compound of a tetrahedron, has its own name: Stella Octangula, which frankly sounds like the name of a science-fiction heroine:

:

What interests me about that shape is it forms six crosses, and if the outermost points are connected, they form a cube.

A tetrahedron has 2 different nets, that is 2 different 2-dimensional shapes that can be folded to obtain a tetrahedron:

 Origami tetrahedron, from WolframMathWorld

A tetrahedron is a three dimensional polytope, a polygon. A 4th-dimensional polytope is a polychoron.The 5-cell, or pentachoron, is the 4D analog of the tetrahedron.  Just as the tetrahedron has four triangular faces, the pentachoron has five tetrahedral faces.

Jason Hise developed that. Visit his webpage http://www.entropygames.net/index.php for more cool stuff.

Applications of tetrahedra, from the  Wikipedia article on Tetrahedron:

### Numerical analysis

In numerical analysis, complicated three-dimensional shapes are commonly broken down into, or approximated by, a polygonal mesh of irregular tetrahedra in the process of setting up the equations for finite element analysis especially in the numerical solution of partial differential equations. These methods have wide applications in practical applications in computational fluid dynamics, aerodynamics, electromagnetic fields, civil engineering, chemical engineering, naval architecture and engineering, and related fields.

### Chemistry

The tetrahedron shape is seen in nature in covalent bonds of molecules. All sp3-hybridized atoms are surrounded by atoms lying in each corner of a tetrahedron. For instance in a methane molecule (CH4) or an ammonium ion (NH4+), four hydrogen atoms surround a central carbon or nitrogen atom with tetrahedral symmetry. For this reason, one of the leading journals in organic chemistry is called Tetrahedron. See also tetrahedral molecular geometry. The central angle between any two vertices of a perfect tetrahedron is $\arccos{\left(-\tfrac{1}{3}\right)}$, or approximately 109.47°).
Water, H2O, also has a tetrahedral structure, with two hydrogen atoms and two lone pairs of electrons around the central oxygen atoms. Its tetrahedral symmetry is not perfect, however, because the lone pairs repel due to their negative charges.
Quaternary phase diagrams in chemistry are represented graphically as tetrahedra.
However, quaternary phase diagrams in communication engineering are represented graphically on a two-dimensional plane.

### Electricity and electronics

If six equal resistors are soldered together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.[8][9]
Since silicon is the most common semiconductor used in solid-state electronics, and silicon has a valence of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on how crystals of silicon form and what shapes they assume.

### Games

Especially in roleplaying, this solid is known as a 4-sided die, one of the more common polyhedral dice, with the number rolled appearing around the bottom or on the top vertex. Some Rubik's Cube-like puzzles are tetrahedral, such as the Pyraminx and Pyramorphix.

### Color space

Tetrahedra are used in color space conversion algorithms specifically for cases in which the luminance axis diagonally segments the color space (e.g. RGB, CMY).[10]

### Contemporary Art

The Austrian artist Martina Schettina created a tetrahedron using fluorescent lamps. It was shown at the light art biennale Austria 2010.[11]
It is used as album artwork, surrounded by black flames on The End of All Things to Come by Mudvayne.

### Geology

The tetrahedral hypothesis, originally published by William Lowthian Green to explain the formation of the Earth,[12] was popular through the early 20th century.[13][14]