A Few Fractals

Following my previous post I have had a number of requests for images of fractals, so here goes.

Fractals are characterised by their property of self-similarity, they are both similar to themselves in different regions and similar to themselves on different length scales, so that an enlargement of a fractal looks the same as the original.

North Britain

A Trip to the Seaside

The notion of fractals was popularized by the Polish mathematician Benoit Mandelbrot (1924-2010) in his book The Fractal Geometry of Nature. Mandelbrot pointed out that many naturally occurring structures are best described in terms of fractals. These include coastlines, trees, clouds, cauliflowers and many other objects. Indeed, any objects that look the same from a distance as when we zoom in to take a closer look. For instance, a small twig often looks just like a miniature tree.

One of Mandelbrot’s favourite examples was the coastline of Britain. As Mandelbrot noted, it is very wiggly. And if we zoom in and take a closer look these wiggles are not ironed out. We just see more smaller wiggles. So that the closer we look the longer the coastline appears to be.

Blowing Your Own Gasket

Mandelbrot was not the first to explore the properties of fractals. Another Polish mathematician Waclaw Sierpinski (1882-1969) is known for a mathematical procedure for generating a fractal structure known as the Sierpinski gasket. Sierpinski’s method of construction was to start with an equilateral triangle, then remove the central triangle to leave three smaller triangles. Each of these triangles is a smaller version of the original triangle. The central triangle is then removed from each of the smaller triangles to leave even smaller equilateral triangles. If this process were continued indefinitely the result would be the Sierpinski gasket, which is composed of a dust of disconnected points.

A similar procedure may be applied to a regular tetrahedron. Removing an octahedron from the centre of the tetrahedron leaves four smaller tetrahedra. Octahedra may then be removed from each of these smaller tetrahedra. Continuing indefinitely produces a tetrahedral Sierpinski gasket, as shown in the animation below.

Julia’s Dream

Another mathematician who took an early interest in fractals was the Frenchman Gaston Julia (1893-1978). He studied the convergence properties of sequences of numbers.These sequences have the following form: take a number c, then for each complex value of x the sequence consists of the terms x(1) = x² + c, x(2) = x(1)² + c, x(3) = x(2)² + c, and so on, each term being equal to the square of the previous term plus c. For some starting numbers x the terms will increase in size indefinitely and the sequence will diverge, whilst for other starting numbers the terms do not increase beyond some finite number. Julia was interested in those values of x that produce sequences that remain finite. These values of x constitute the Julia set of c.

This might sound rather technical, but it is the sort of procedure that a computer is ideally suited to performing and the end results look rather attractive. It is only since computers became commonplace that mathematicians have been able to explore these objects in detail.

By generating a sequence of Julia sets for values of the parameter c corresponding to points on a closed loop it is possible to create animations such as the ones shown here. I originally produced these animations some years ago for the Fractal Geometry section of my CD-ROM Art and Mathematics.

The animations illustrate just how complicated the structure of the Julia sets can be and also how their structure is dramatically dependent on the value of the constant c. The Julia sets fall into two classes: they are either completely connected together or they consist of a dust of isolated points.

Earlier this year on 4th July researchers working at the Large Hadron Collider made their epoch-making announcement that they had discovered a particle that looks very much like the long-awaited Higgs boson. It was clear that something new had turned up in the detectors of the LHC, but no-one could be absolutely certain that it really was the much expected Higgs boson.

The Best Theory That We Have!

The CMS detector in the Large Hadron Collider during construction. (Copyright CERN, Geneva.)

The best theory that we have to describe all the particles and forces observed in particle accelerators is known as the Standard Model. (This isn’t a great name, but it reflects the success of the theory and the confidence that physicists place in it.) According to the Standard Model the Higgs field is responsible for breaking the ‘electroweak’ force into two forces that we observe as the electromagnetic and weak forces. It does this by giving mass to the three particles, the W-minus, W-plus and Z-nought bosons that are responsible for producing the weak force, while leaving the photon – which is responsible for the electromagnetic force – massless. This makes the weak force feeble and very short range, while the electromagnetic force remains powerful and long range.

The Multi-tasking Higgs Boson

But the Standard Model requires more of the Higgs field. According to the Standard Model the Higgs field is also responsible for giving mass to other particles, the particles from which matter is formed such as electrons and quarks.

A Bit of Detective Work

The Higgs boson cannot be observed directly in the detectors of the LHC. What the detectors see are the particles that the Higgs decays into. It then takes a bit of detective work to interpret these fingerprints. From the original announcement in July it seemed pretty clear that the new particle was doing the first part of the job of the Higgs, that is it was decaying into Ws and Zs and was therefore responsible for producing their mass (and breaking the electroweak force).

Mural on the building above the ATLAS detector at the Large Hadron Collider. Mural by Josef Kristofoletti. (Copyright CERN, Geneva.)

What was not so certain was whether it was also decaying into quarks and tauons (heavier relatives of the electron) at the expected rate. And was therefore also responsible for producing the mass of the electrons and quarks. The decays into tauons would be much rarer, so more datawas needed to confirm the predictions.

Further Progress

A further announcement about progress at the Large Hadron Collider was made earlier this week. The detectors of the LHC have now collected twice the data that they had by the time of the original announcement and it seems as though the Standard Model predictions are holding up and the new particle really is the Standard Model Higgs boson. This is great news of course, unless you are a physicist looking for something new and surprising. (But you can’t please everyone!)

Deeper Connections

We know that the Standard Model cannot be the final answer. It is actually formed of two theories coupled together: one describes the electroweak force, the other, known as quantum chromodynamics, describes the strong force which holds quarks together to form protons and neutrons, and holds protons and neutrons together to form atomic nuclei. There must be some deeper connection between the electroweak force and the strong force that is missing from the Standard Model.

The Standard Model also fails to explain some of the most significant large scale features of the Universe, such as the fact that most of the matter that it contains seems to be formed of an unknown substance that is referred to as ‘dark matter’.

Higgs Force: Cosmic Symmetry Shattered by Nicholas Mee

An Even Better Theory

So the fact that the Standard Model predictions are holding up is a major triumph for physics, but what physicists really want are some surprises that will provide clues that will lead to a new and even better theory.

I’m sure that we won’t have to wait long before the Large Hadron Collider produces something really unexpected.

More Information

There is, of course, much more information about the search for the Higgs boson in my book Higgs Force:
http://quantumwavepublishing.com/higgs-force/

Inspirational Games

Insp
irational Games Arithmetical games are a great way to sharpen up your arithmetic. My Grandad taught me two classic games that gave me a familiarity with numbers before I was even old enough to go to school. These two games are Fives and Threes which is a dominoes game and Cribbage which is a card game.

Fives and Threes
Fives and Threes is played with a peg board to keep track of the score of each player. It should be played with a set of ‘Double Nine’ dominoes, as this offers a much more interesting range of number possibilities. As usual with domino games, players take turns to build a chain of dominoes. But the distinctive feature of Fives and Threes is that players score points depending on the values of the exposed ends of the domino chain. The amount scored is equal to the number of times that the sum of the two ends can be exactly divided by three or five. For instance, if one end of the chain shows a 2 and the other end shows a 4, then the total is 6, which is exactly divisible by 3 twice, so the person who played the previous domino scores 2. The highest scoring totals are those that are simultaneously divisible by 5 and 3. For instance, if one end of the chain is 7 and the other end is 8, this produces a total of 15, which is divisible by both 5 and 3. It is divisible by 5 three times and it is divisible by 3 five times, so this gives a score of 3 + 5 = 8.

The full rules of Fives and Threes are given on the following website:
http://www.pagat.com/tile/wdom/fives_and_threes.html

Cribbage
The card game Cribbage is also played with a peg board to keep track of the scores. Points are scored for card combinations that add up to fifteen, and for pairs, triples, quadruples, runs and flushes. As well as the obvious arithmetical task of finding cards whose values sum to 15, Cribbage is of interest because of the importance that combinations and permutations have in the game. For instance, a pair of numbers with the same value, say two 7s, would be worth two points. A triple, say three 7s, would be worth six points, because there are three ways to choose a pair of 7s from the triple, and three times two (the value of a pair) is six. A quadruple of four 7s is worth twelve points, because this gives six ways to choose a pair of 7s, and six times two is twelve.

Points are also awarded for every combination of cards that sum to 15. For instance, a hand formed of a 6, two 7s and two 8s, can be combined to form 15 in four ways, because each of the 7s can be added to each of the 8s to form 15. Similarly points are scored for every combination of cards that produces a run of three or more cards. For instance, a 6, two 7s and two 8s, would produce four runs of three cards.

This may sound a bit complicated if you are unfamiliar with Cribbage, but it is quite straightforward really and definitely worth investigating. The full rules of Cribbage are given on the following website:
http://www.pagat.com/adders/crib6.html

According to John Aubrey, Cribbage was invented by the aristocratic English poet Sir John Suckling early in the 17th century.

http://en.wikipedia.org/wiki/John_Suckling_(poet)

Both Fives and Threes and Cribbage are traditional British pub games, but they are great games nonetheless. Cribbage remains my favourite card game.

The Richmomachia board at the start of a game.

RithmomachiaThe most elaborate of all arithmetic games must be Rithmomachia or The Philosophers’ Game. The name Rithmomachia means ‘Battle of the Numbers’. Descriptions of the game survive in manuscripts from as long ago as the 11th and 12th centuries, when it seems to have been played in Benedictine monasteries. Rithmomachia is like an arithmetical relative of Chess. Each player has a number of circular, triangular and square pieces, as well as a single pyramid. These pieces have numerical values. Like Chess, play involves the capture of the pieces belonging to the opponent, but unlike Chess the success of an attack depends on the number combinations of the pieces.

The game is reputed to have been invented as a method of teaching Pythagorean numerology, as set out in The Consolation of Philosophy written by the fifth century poet Boethius – a book that was very highly esteemed throughout the European Middle Ages. Rithmomachia was highly regarded by leading late medieval British philosophers and mystics, such as Roger Bacon, Thomas More and John Dee.

More information about Rithmomachia and a description of the rules are given on the following website:
http://jducoeur.org/game-hist/mebben.ryth.html

The Nubble! board.

Nubble!
The greatest arithmetic game of modern times is Nubble! It was invented in 1994 by Edgar Fineberg and Jack Berkovi and was originally published as a board game by Dorling Kindersley under the name Number Quest. The Nubble! board is composed of 100 hexagons. Players take turns to throw four dice and then combine the scores of all four dice by addition, subtraction, division and multiplication to produce a number between 1 and 100. They then place a counter on the corresponding hexagon and points are scored depending on the zone in which the hexagon lies. Bonus points are also available if the player scores a “Nubble!”, which is achieved by forming a little triangle of three adjacent counters. Even more bonus points are awarded for a “Double Nubble!”, which is produced by placing a counter on a prime number to form a Nubble!

Nubble! was the only board game to receive a Millennium Product Award from the Design Council.

The Nubble! software is now available from Quantum Wave Publishing. Find out more by clicking the following link:
http://quantumwavepublishing.com/nubble/

What on Earth is a Boson?

What on Earth is a Boson?

Quantum Waves and the Rosetta Stone

Quantum Waves and the Rosetta Stone

1 Light Year

1 Light Year

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RedBull’s Stratos Stunt

Posted on October 16, 2012 by asteitel

Baumgartner, covered in RedBull logos, begins his fall at just over 128,000 feet. Credit: Handout/Getty Images via The Guardian

According to YouTube, eight million people watched Felix Baumgartner’s high altitude jump on Sunday morning. It was exciting and death-defying, but at the end of the day it was a just an elaborate publicity stunt that will likely see RedBull sales skyrocket this month. But I’d argue that the event wasn’tentirely a success from a publicity standpoint. RedBull, who sponsored the jump, wasted an incredible opportunity. It had  an eight million person audience captivated, but did nothing to teach that audience about the context behind Baumgartner’s jump. Joe Kittinger’s 1960 jump was amazing, the heritage behind these types of tests is fascinating, but without any context the audience just saw a daredevil break a record for record-breaking’s sake.

I realize I sound like an irritated historian, but I also have a background (albeit a brief one) in publicity. Not taking advantage of an opportunity to teach eight million people a few awesome things about science is a terrible waste, from an historian’s standpoint and a public relation’s standpoint.

What ever else he may be, Baumgartner is definitely agood face for RedBull. Credit: Rex

A little background first. Austrian-born Baumgartner started skydiving at 16. He perfected the art and in 1988 began performing skydiving exhibitions for Red Bull. His adventurous spirit and RedBull’s out-of-the-box thinking meshed well, sparking a now decades-long collaboration. The idea for a free fall from the stratosphere, a planned altitude of 120,000 feet, was conceived in 2005. It was finally named The RedBull Stratos project, and its goal was defined as transcending “human limits that have existed for 50 years.”

Ostensibly, the jump was designed to expand the boundaries of human flight. More concrete goals listed on the project’s website include: developing new spacesuits with enhanced mobility and visual clarity to assist in “passenger/crew exit from space;” developing protocols for exposure to high altitude and high acceleration environments; exploring the effects of supersonic acceleration and deceleration on the human body; and testing the latest innovations in parachute systems.

The International Space Station, which you can’t jump out of. Credit: NASA

It’s not entirely clear what applications this data would have, like the research on “passenger/crew exit from space.” The morning of the jump, people asked me whether the point was to prove that astronauts could jump from the International Space Station in an emergency. It wasn’t. Baumgartner’s 128,000 foot altitude (he overshot his mark) is only about 24 miles; the ISS orbits at an altitude of about 200 miles. Not to mention the astronauts on the ISS are weightless because they’re falling around the Earth at the same rate as the station, and that wouldn’t change if they stepped outside. It’s also unclear what other high altitude/high acceleration and supersonic environments in which people would find themselves that we need data on survival. Yes, there may have been some interesting data gathered from the jump, but it’s not enough to classify the stunt as any kind of research program.

But what bothered me the most is how RedBull presented the jump. Saying that the Stratos project was designed to “transcend human limits that have existed for 50 years” is a good tagline but it’s vague. Jumping from 24 miles doesn’t push human limits so much as technological limits. Technology kept Baumgartner alive during his ascent, protected him from the harsh environment during the fall, and slowed him to a soft landing. The other thing that stands out in the tagline is its implication that we haven’t learned anything about surviving in these types of extreme environments since 1962. Test pilots and astronauts in the 1960s were subjected to excessive G-forces, relied on intricate life support systems throughout missions, and were spared exposure to the vacuum of space by spacesuits.

A schematic showing the layers of Earth’s atmosphere. The stratosphere isn’t quite space. Credit: NASA

Which brings up another problem with RedBull’s promotion of the Stratos jump. It was touted as being a jump from space, but 24 miles isn’t space. There’s no clear limit where the atmosphere ends and space begins, but the general consensus is that it’s around the 62 mile mark. NASA, which was established to run the space game in 1958, has awarded astronaut wings to pilots who’ve flown higher than 50 miles. Calling the Stratos jump a jump from space is just not true, and unfortunately with eight million people watching those eight million people now have a very wrong idea about where space starts.

This was far from the only misinformation associated with the event. RedBull did a terrible job at presenting Kittinger’s 1960 jump. A real shame ,especially since Kittinger was Baumgartner’s capcom. From the RedBull Stratos website:

Joe’s record jump from 102,800 ft in 1960 was during a time when no one knew if a human could survive a jump from the edge of space… Although researching extremes was part of the program’s goals, setting records wasn’t the mission’s purpose. Joe ascended in [a] helium balloon launched from the back of a truck. He wore a pressurized suit on the way up in an open, unpressurized gondola. Scientific data captured from Joe’s jump was shared with U.S. research personnel for development of the space program.

This description isn’t just wrong, it completely ignores the history behind Kittinger’s jump.

Captain Mel Apt in the cockpit of the X-2 in 1956. He became the first man to fly faster than Mach 3 in this aircraft, but lost control at high altitude. He ejected, but the complicated system failed and he was killed on this record flight. The U.S. Air Force needed to stop this from happening again. Credit: United States Air Force

In the 1960s, pilots were pushing the envelope of supersonic flight at high altitudes. But this was a dangerous approach. While it’s easy to fly fast in the thin upper atmosphere it’s harder to control an aircraft. With no air for control surfaces to push against, aircraft tend to tumble, and when aircraft tumble pilots tend to eject. Tests with dummies showed that when falling from high altitudes, human bodies tended to get into a flat spin. It would be like rolling down a hill really fast but without the hill, and the G-forces would certainly be fatal. The Air Force needed a way to stabilize a pilot from a high altitude ejection, and Francis F. Beaupre had a sequential parachute that would do just that. Kittinger jumped from 102,800 feet in 1960 as part of Project Excelsior to prove that Beaupre’s parachute would work. It did, the Air Force had data and a healthy Kittinger as evidence, and the project ended. There was no live video of his jump. He was a Captain in the Air Force, and he jumped from 102,800 feet for Captain’s pay to complete a mission.

The full story behind Kittinger’s jump is a fascinating one. It pulls together classic themes like 1960s test pilots’ egos, their relationships with their aircraft, the push from atmospheric flight to spaceflight, and the era where men were probing unknowns because they were unknown.

Joe Kittinger in his U.S. Air Force days. Credit: United States Air Force

During Baumgartner’s more than two hour long ascent to jump altitude, RedBull could have told Kittinger’s story. The announcer could have talked about the technology keeping Baumgartner alive, what made his suit different or special, told us how he was able to break the sound barrier in a free fall, talked about problems like aerodynamic heating in atmospheric entry. Instead, RedBull held an audience captive and offered them almost nothing but shots of Baumgartner in a suit and Kittinger at the capcom console. Even when the announcer talked about the possibility of Baumgarner entering a spin during his fall, he failed to mention the parallel that Kittinger had proved the graduated parachute system that stabilized a pilot’s fall. He didn’t even mention that Baumgartner’s supersonic jump came on the 65th anniversary of Chuck Yeager’s first supersonic flight.

RedBull Stratos was an incredible opportunity to teach a huge audience about the past and future exploration of high altitudes and space. Having a scientist or historian narrating the jump would have brought a level of prestige to the event. It could have been less of a publicity stunt and more of an event designed to return scientific data that just happened to be sponsored by a corporation.

I can’t help but think this Stratos jump could have been more powerful and interesting had we learned the context behind the mission. In the end, I have to wonder how much we’re gaining if the public is excited by space exploration but doesn’t understand why it matters or the technology behind it.

The view as Kittinger jumped from the gondola at 102,800 feet in 1960. It’s a great picture, made more powerful knowing what Kittinger was out to prove and how rudimentary (by today’s standards) the precautions were. A leak in his suit caused his right hand to swell painfully during the ascent, but he didn’t say anything for fear the jump would be cancelled. Credit: United States Air Force

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This entry was posted in History of Space Science and tagged Felix Baumgartner, Free Fall, Joe Kittinger, Jump, Mel Apt, NASA, Project Excelsior, RedBull, RedBull Stratos, Skydive, Stratosphere, Stunt, US Air Force. Bookmark thepermalink.

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