Month: August 2016

How Long Would It Take You To Fall Through Earth?

Just like when you were a kid and you were digging on the beach, imagine if there was a tunnel directly through Earth. If you jumped in, how long would it take to reach the other side?

Life Noggin has created this 3-minute animation showing all the mathematics and physics you need to answer this question that has probably bothered you since you were four.

To make it a bit easier, lets assume the same density all throughout, the world is perfectly spherical, and the hole has no air in it. Even then, theres that annoying little fundamental force known as gravity. When we start our fall, the mass of Earth pulls us towards its center, but what happens when we reach the center and go beyond it?

Check out the video for all the answers.

Read more: http://www.iflscience.com/physics/how-long-would-it-take-you-to-fall-through-earth/

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Coding on tape – computer science A-level 1970s style – BBC News

Image copyright The National Museum of Computing  
Few schools had their own computers in the early 1970s – but by the end of the decade many began to invest in models such as the Elliott 903

The chances of walking into a UK school without any computer equipment in today’s world are practically zero.

But 45 years ago things were very different.

Staff at the AQA exam board have been trawling their archive and have found some early 1970s computer science papers, which highlight how much the subject has changed over the intervening period.

Back then, schools offering computer science A-level had to prove they had access to a computer, according to a syllabus from the time.

In 1970 computers were a rarity and pupils would have to visit machines in nearby universities or businesses, said AQA’s computing qualifications manager Steven Kenny, because the cost of a school owning one was “prohibitive”.

“They cost tens of thousands of pounds and were filing cabinet sized,” he explained.

Paper tape

Students would write programmes in longhand and if they were lucky, their school would have the facilities to punch the code on to paper tapes.

Otherwise the longhand notes would have to be sent off to be converted on to tape.

The students would then take the code with them when they visited the computer.

“They would spool the paper on to the computer and the code would appear on the screen if they had got it right. If it was wrong they would get an error message,” said Mr Kenny.

This meant they would have to take it back to school, work out was wrong and correct it, he explained.

“It was often the same even at universities in those days.”

Image copyright Picasa 
Computer code was punched on to paper tape and then loaded on to the computer to be read
Image copyright AQA
Image caption Schools had to prove they had access to a computer
Image copyright AQA
  Parts of the paper seem quaint by today’s standards

Kevin Murrell, co-founder of The National Museum of Computing, said even a comma or a full stop out of place could mean a programme would fail once it was put into the computer.

He described himself as “mortified” by the difficulty of the mathematics required by the 1971 A-level.

When he showed the paper to his staff, they “went white as a sheet”, he told the BBC.

The paper required candidates to break everything down into binary or “machine code”, which is “very hard work and not something that anyone would be asked to do today” he said.

“We write in languages that are akin to English which are converted into binary by the machine. For 99.9% of programmers, maths isn’t an issue at all because modern computers do so much more.”

In the 1970s machines were slow and students’ access was so limited they had to be very precise about the programmes they wrote.

Today’s programmers can afford to be less precise as “the sheer brute force of modern computers gets you through”.

Mr Murrell did O-level computer science in 1976 using a keyboard and printer connected by phone to an Open University computer.

“You would key in your programme and it would run remotely at the OU. I can’t tell you how exciting that was!”

Another generation

The price of computers gradually dropped with a Research Machines computer costing around 3,000 by the mid 1970s and a Commodore PET around 1,000 after that.

Another generation of coders learned their craft on machines like the Sinclair Spectrum and the BBC Micro which appeared in the early 1980s.

Mr Kenny remembers the arrival of a computer at his inner city comprehensive school in 1979.

“Pupils had very limited access – only a few were allowed to use it.”

All this had an effect on the exam questions.

“Because people didn’t have access to computers, a lot of it had to be focused on theory.

“There was a big focus on logical and computational thinking, using flow charts and algorithms to break instructions down into the smallest logical steps for a computer to follow.

“The difference now is that there is a lot more creativity involved: for example designing apps for use in daily life.”

However, one surprise was how many of the questions are still relevant, said Mr Kenny.

“About half the questions could probably be put into an A-level paper today with a bit more context.”

The big differences were in questions focused on the practical use of computers – some of which now seem quaint.

For example candidates were asked to “Write an essay on the use of computers in one of the following (a) airlines (b) banking (c) local government or (d) wholesale trade”.

Another question required them to prepare a flow chart to record 12,000 ticket sales between four destinations on the same railway line over a 30-day period.

“We wouldn’t phrase this the same way or use the same example but some of the same processes would be needed by programmers today,” said Mr Murrell.

Mr Kenny suggested today’s equivalent question would be about “the internet of things”, where, for example, a computerised fridge can detect when groceries are running low and order a delivery without the householder being involved.

‘Geeky’

By the early 1980s people began to buy home computers and Mr Kenny admitted his younger self used to wonder why people would use them.

Image copyright Picasa
  A 1980s BBC Micro in use at The National Museum of Computing

“They would have been game-players or enthusiastic coders – much more geeky than now,” he suggests before adding that some people might have used them for “numbers and data”, particularly home finances.

He adds: “It’s easy for us to answer that question today because computers are everywhere. You have to remember that in 1971 not many people had even seen a computer, let alone used one.”

Read more: http://www.bbc.co.uk/news/education-35890450

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Mathmetician solves puzzle of parking lots

 (iStock)

A Mathematician has discovered a simple trick which could revolutionize car parking.

Professor David Percy, of Salford University, was frustrated by the poor geometry of car parks, which generally feature parking spaces arranged at a 90 degree angle to the access lane.

He performed some simple math and found out that one tweak could make it much easier for motorists to maneuver their motor into the space whilst simultaneously allowing more automobiles to fit into a car park.

All planners need to do is place the bays at a 45 degree angle, which cuts down the turning circle required and therefore needs a smaller access lane, freeing up more space for parking.

Professor Percys eureka moment came when his universitys car park was repainted, sticking to exactly the same layout.

This traditional conformity set me thinking, he wrote in Mathematics Today.

To understand his plan, you need to look at the space required to fit a car into a bay thats angled at 90 degrees.

Maneuvering a car into traditional bays requires a wide access road, whilst anyone the 45 degree bays will be able to swing in their car without needing as much space.

Instead of rectangular parking bays I figured if they are diagonal you might save space and fit more cars into a car park, he added.

If his recommendations are implemented in a car park which can currently fit 500 cars, another 119 should be able to fit into the same space.

For a 45 degree bay angle it was a 23 percent saving, he continued,

For 36 degrees it was 34 percent, but the difference is marginal and its easier to draw lines at 45 degrees.

His plans only work in larger car parks, because the useless space at the edge of bays negates the effects of the space-saving lanes.

More auto news from Sun Motors

Read more: http://www.foxnews.com/leisure/2016/08/25/mathmetician-solves-puzzle-parking-lots/

See If You Can Solve These 6 Fiendish Brain-Teasers Written By NSA Employees

Sure, between thespying on millions of Americans and thesitting on critical security vulnerabilities for years, the National Security Agency (NSA) has a pretty bad reputation.

But have you heard about itsbrain-teasers?

The US spy agencyemploys some of America’s best and brightest as spooks and code-breakers, so it’s no surprise that its employees might have an interest in riddles and puzzles.

Every month, the NSA publishes on its website a brain-teaser written by an employee that members of the public can try their hand at.

One month it’s a maths challenge created by an applied research mathematician; the next it’s a logic puzzle by a systems engineer. They’re all published in what the NSA calls its “Puzzle Periodical.”

“Intelligence. It’s the ability to think abstractly. Challenge the unknown. Solve the impossible. NSA employees work on some of the world’s most demanding and exhilarating high-tech engineering challenges. Applying complex algorithms and expressing difficult cryptographic problems in terms of mathematics is part of the work NSA employees do every day,” the NSA says on its website.

We’ve rounded up a sixof the most interesting brain-teasers below. So take a read, and see if you can out-smart the NSA’s most fiendishriddlers!

1. Here’s a relatively easy one to start off with, from July 2016:

Submitted by Sean A., NSA Applied Mathematician

content-1471883922-heres-a-relatively-eaOn a rainy summer day, brothers Dylan and Austin spend the day playing games and competing for prizes as their grandfather watches nearby. After winning two chess matches, three straight hands of poker and five rounds of ping-pong, Austin decides to challenge his brother, Dylan, to a final winner-take-all competition. Dylan clears the kitchen table and Austin grabs an old coffee can of quarters that their dad keeps on the counter.

The game seems simple as explained by Austin. The brothers take turns placing a quarter flatly on the top of the square kitchen table. Whoever is the first one to not find a space on his turn loses. The loser has to give his brother tonights dessert. Right before the game begins, Austin arrogantly asks Dylan, Do you want to go first or second?

Dylan turns to his grandfather for advice. The grandfather knows that Dylan is tired of losing every game to his brother. What does he whisper to Dylan?

Image credit:MoneyBlogNewz/Flickr (CC)

And here’s the solution:

Dylan should go first. By doing this, Dylan can guarantee a win by playing to a deliberate strategy. On his first turn, he can place a quarter right on the center of the table. Because the table is symmetric, whenever Austin places a quarter on the table, Dylan simply “mirrors” his brothers placement around the center quarter when it is his turn. For example, if Austin places a quarter near a corner of the table, Dylan can place one on the opposite corner. This strategy ensures that even when Austin finds an open space, so can Dylan. As a result, Dylan gains victory, since Austin will run out of free space first!

2. This one, from June 2016, requires a bit more math.

Submitted by Robert B., NSA Applied Mathematician

content-1471884144-this-one-from-june-20Following their latest trip, the 13 pirates of the ship, SIGINTIA, gather at their favorite tavern to discuss how to divvy up their plunder of gold coins. After much debate, Captain Code Breaker says, Argggg, it must be evenly distributed amongst all of us. Argggg. Hence, the captain begins to pass out the coins one by one as each pirate anxiously awaits her reward. However, when the captain gets close to the end of the pile, she realizes there are three extra coins.

After a brief silence, one of the pirates says, I deserve an extra coin because I loaded the ship while the rest of you slept. Another pirate states, Well, I should have an extra coin because I did all the cooking. Eventually, a brawl ensues over who should get the remaining three coins. The tavern keeper, annoyed by the chaos, kicks out a pirate who has broken a table and who is forced to return her coins. Then the tavern owner yells, Keep the peace or all of you must go!

The pirates return to their seats and the captain, left with only 12 total pirates, continues to distribute the coins – one for you, one for you. Now, as the pile is almost depleted, she realizes that there are five extra coins. Immediately, the pirates again argue over the five extra coins. The captain, fearing that they will be kicked out, grabs the angriest pirate and ushers her out of the tavern with no compensation. With only 11 pirates left, she resumes distribution. As the pile nears depletion, she sees that there wont be any extra coins. The captain breathes a sigh of relief. No arguments occur and everyone goes to bed in peace.

If there were less than 1,000 coins, how many did the pirates have to divvy up?

Image credit: Disney

The answer is 341.

There are actually infinite answers to the problem, but only one number if the answer is under 1,000. This puzzle is an example of modular arithmetic and the Chinese Remainder Theorem.

The smallest solution under 1,000 for this problem is 341 coins, and the answer is found by working backwards. To find it, we first note that with 11 pirates the coins divided evenly; hence, the number of coins is in the list:

11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143

What happens if we take these numbers and divide them among 12 pirates? How many coins would be left over? Well, we want 5 coins to be left over after dividing by 12. Hence, we reduce the list above to:

77, 209, 341, 473

These numbers divide by 11 evenly and have 5 left over when divided by 12. Now we take these remaining numbers and divide them by 13 until we find the number that gives 3 extra coins left over. Hence, 341 coins.

3. Here’s a logic puzzle from August 2015. You need to figure two things out.

Puzzle created by Roger B., Cryptanalytic Mathematician, NSA

content-1471884232-heres-a-logic-puzzle-Nadine is having a party, and she has invited three friends Aaron, Doug, and Maura. The three of them make the following statements on the days leading up to the party:

Two Days Before the Party:
Aaron: Doug is going to the party.
Doug: Maura is not going to the party.
Maura: Aaron will go to the party if, and only if, I do.

The Day Before the Party:
Aaron: Maura will go to the party if, and only if, I don’t go.
Doug: An even number out of the three of us are going to the party.
Maura: Aaron is going to the party.

The Day of the Party:
Aaron: It is not yet 2018.
Doug: Aaron will go to the party if, and only if, I do.
Maura: At least one of the three of us is not going to the party.

Nadine also knows that out of Aaron, Doug, and Maura:
One of them never lies.
A different one of them lies on days of the month that are divisible by 2, but is otherwise truthful.
The remaining one of them lies on days of the month that are divisible by 3, but is otherwise truthful.

(1) Can you figure out who is going to attend?
(2) Can you figure out on what day, month, and year the party will be held, assuming it takes place in the future?

Image credit:YouTube/blackboxberlin

Stuck? The answers are Doug and Aaron, and March 1, 2016. Here’s how you get there …

(1) Attendees solution:

The first step is to realize that the date rules imply that nobody may lie on two consecutive days.

If Maura’s third statement is false, then everyone is going to the party and Doug’s first two statements are lies, which is impossible. Therefore Maura’s third statement is true, and at least one person is not going to the party.

If Doug’s second statement is false, then (from the above) exactly one person is going to the party. This can’t happen without making Doug’s first or third statement also false, which would have Doug lying on two consecutive daysan impossibility. Therefore Doug’s second statement is true, and there are an even number of people going to the party.

If Doug is not going to the party, then Aaron’s first statement is false, so his second statement must be true. This would result in an odd number of people at the party, which we have shown is not the case. Therefore Doug is going to the party.

Since an even number of people are going to the party, only one of Aaron or Maura is going, making Maura’s first statement false. Therefore her second statement must be true, and Aaron is going to the party while Maura is not. The attendees are Doug and Aaron.

(2) Date solution:

Having established the attendees as Doug and Aaron, now we know that the only lies are Maura’s first statement and potentially Aaron’s third statement.

If the three days do not cross a month boundary, then either the second date or both the first and third dates would be divisible by 2, but nobody is available to lie with that pattern. Therefore either the second or third day is the first of a month.

If the day before the party was the first of a month, then the day of the party is the second of a month; Aaron would have to be the one who lies on dates divisible by 2. Then, for Maura but not Aaron to lie on the first day, it would have to be divisible by 3 but not 2. This is never true of the last day of a month. Therefore, the day before the party is not the first of a month, so the day of the party itself must be the first day of a month.

This makes the day before the party the last day of a month, and since nobody lies on that day it must not be divisible by 2.

This means two days before the party, the date is divisible by 2, so it must not also be divisible by 3 or there would be two liars on that day. The only way this can happen two days before the end of a month is when that day is February 28 of a leap year.

Since nobody lies on the first of a month, Aaron’s third statement is true and it is not yet 2018.

Finally, since the only leap year before 2018 is 2016, we conclude that the party is being held onMarch 1, 2016.

4. This one (April 2016) is simpler but not any easier.

Puzzle created by Andy F., Applied Research Mathematician, NSA

content-1471884342-this-one-april-2016-iMel has four weights. He weighs them two at a time in all possible pairs and finds that his pairs of weights total 6, 8, 10, 12, 14, and 16 pounds. How much do they each weigh individually?

Note: There is not one unique answer to this problem, but there is a finite number of solutions.

Image credit:Doug Pensinger/Getty Images

And the answer is …

There are exactly two possible answers: Mel’s weights can be 1, 5, 7, and 9 pounds, or they can be 2, 4, 6, and 10 pounds. No other combinations are possible.

Explanation

Let the weights be a, b, c, and d, sorted such that a < b < c < d. We can chain inequalities to get a + b < a + c < a + d, b + c < b + d < c + d. Thus, a + b = 6, a + c = 8, b + d = 14, and c + d = 16. But we don’t know if a + d = 10 and b + c = 12 or the other way around. This is how we get two solutions. If a + d = 10, we get 1, 5, 7, and 9; if b + c = 10, we get 2, 4, 6, and 10.

More on the Problem

Where this problem really gets weird is that the number of solutions depends on the number of weights. For example, if Mel has three weights and knows the weight of all possible pairs, then there is only one possible solution for the individual weights. The same is true if he has five weights.

But now suppose that Mel has eight weights, and the sums of pairs are 8, 10, 12, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 36, 38, and 40. Now what are the individual weights?

This time, there are three solutions:

  • 1, 7, 9, 11, 13, 15, 17, 23
  • 2, 6, 8, 10, 14, 16, 18, 22
  • 3, 5, 7, 11, 13, 17, 19, 21

5. October 2015 was another logic puzzler, with a school theme. Take a look:

Puzzle created by Ben E., Applied Research Mathematician, NSA

Kurt, a math professor, has to leave for a conference. At the airport, he realizes he forgot to find a substitute for the class he was teaching today! Before shutting his computer off for the flight, he sends an email: “Can one of you cover my class today? I’ll bake a pie for whomever can do it.” He sends the email to Julia, Michael, and Mary Ellen, his three closest friends in the math department, and boards the plane.

As Kurt is well-known for his delicious pies, Julia, Michael, and Mary Ellen are each eager to substitute for him. Julia, as department chair, knows which class Kurt had to teach, but she doesn’t know the time or building. Michael plays racquetball with Kurt so he knows what time Kurt teaches, but not the class or building. Mary Ellen helped Kurt secure a special projector for his class, so she knows what building Kurt’s class is in, but not the actual class or the time.

Julia, Michael, and Mary Ellen get together to figure out which class it is, and they all agree that the first person to figure out which class it is gets to teach it (and get Kurt’s pie). Unfortunately the college’s servers are down, so Julia brings a master list of all math classes taught that day. After crossing off each of their own classes, they are left with the following possibilities:

  • Calc 1 at 9 in North Hall
  • Calc 2 at noon in West Hall
  • Calc 1 at 3 in West Hall
  • Calc 1 at 10 in East Hall
  • Calc 2 at 10 in North Hall
  • Calc 1 at 10 in South Hall
  • Calc 1 at 10 in North Hall
  • Calc 2 at 11 in East Hall
  • Calc 3 at noon in West Hall
  • Calc 2 at noon in South Hall

After looking the list over, Julia says, “Does anyone know which class it is?” Michael and Mary Ellen immediately respond, “Well, you don’t.” Julia asks, “Do you?” Michael and Mary Ellen both shake their heads. Julia then smiles and says, “I do now. I hope he bakes me a chocolate peanut butter pie.”

Which class does Kurt need a substitute for?

The answer is Calc 2 at 10 in North Hall. But why?

1) Since Julia only knows the class name, the only way she could immediately know is if it was Calc 3. Since Michael and Mary Ellen both know that Julia doesn’t know, that means they know the class isn’t Calc 3. Since Michael only knows the time, that means the class can’t be at noon. Because Mary Ellen only knows the building, that means the building can’t be West Hall. That leaves only the following possibilities:

  • Calc 1 at 9 in North Hall
  • Calc 1 at 10 in East Hall
  • Calc 2 at 10 in North Hall
  • Calc 1 at 10 in South Hall
  • Calc 1 at 10 in North Hall
  • Calc 2 at 11 in East Hall

2) Since Michael doesn’t know which class it is, that means the time can’t be 9 or 11. Since Mary Ellen doesn’t know either, the class can’t be in South Hall. That leaves only three possibilities:

  • Calc 1 at 10 in East Hall
  • Calc 2 at 10 in North Hall
  • Calc 1 at 10 in North Hall

3) At this point, Julia now says she knows the answer. Since there are two Calc 1 classes, it must be that Julia knows the class is Calc 2. Thus, the class is Calc 2 at 10 in North Hall.

6. Last one: Can you figure out Charlie’s birthday? (July 2015)

Puzzle created by Stephen C., Applied Research Mathematician, NSA

After observing Albert and Bernard determine Cheryl’s birthday, Charlie decides he wants to play. He presents a list of 14 possible dates for his birthday to Albert, Bernard and Cheryl.

  • Apr 14, 1999
  • Feb 19, 2000
  • Mar 14, 2000
  • Mar 15, 2000
  • Apr 16, 2000
  • Apr 15, 2000
  • Feb 15, 2001
  • Mar 15, 2001
  • Apr 14, 2001
  • Apr 16, 2001
  • May 14, 2001
  • May 16, 2001
  • May 17, 2001
  • Feb 17, 2002

He then announces that he is going to tell Albert the month, Bernard the day, and Cheryl the year.

After he tells them, Albert says, “I don’t know Charlie’s birthday, but neither does Bernard.”

Bernard then says, “That is true, but Cheryl also does not know Charlie’s birthday.”

Cheryl says, “Yes and Albert still has not figured out Charlie’s birthday.”

Bernard then replies, “Well, now I know his birthday.”

At this point, Albert says, “Yes, we all know it now.”

What is Charlie’s birthday?

Image credit:Will Clayton/Flickr (CC)

The answer is Apr 16, 2000, and the rationale is …

When Albert claims that Bernard does not know Charlie’s birthday, he is saying that he knows that the correct day occurs more than once in the list. In other words, he is saying that Charlie’s birthday is not Feb 19, 2000, and the only way he could know that is that he knows that the month is not February.

So by making this claim Albert has reduced the list for everyone to:

  • Apr 14, 1999
  • Mar 14, 2000
  • Mar 15, 2000
  • Apr 16, 2000
  • Apr 15, 2000
  • Mar 15, 2001
  • Apr 14, 2001
  • Apr 16, 2001
  • May 14, 2001
  • May 16, 2001
  • May 17, 2001

When Bernard says that it is true that he does not know Charlie’s birthday, it tells everyone that even with the restricted list the correct day occurs more than once in the list. So everyone can thus eliminate May 17, 2001.

Furthermore, the claim that Cheryl also does not know Charlie’s birthday is a claim that Bernard knows that the year occurs more than once on the remaining list.

This rules out Apr 14, 1999, and since Bernard could only rule this out by knowing that the day was not 14, everyone can further reduce the list to:

  • Mar 15, 2000
  • Apr 16, 2000
  • Apr 15, 2000
  • Mar 15, 2001
  • Apr 16, 2001
  • May 16, 2001

When Cheryl says that Albert still has not figured out Charlie’s birthday, she is telling everyone that given the new list the month occurs more than once and thus rules out May 16, 2001. This tells everyone that Cheryl knows that the year is not 2001 and the list can be reduced to:

  • Mar 15, 2000
  • Apr 16, 2000
  • Apr 15, 2000

When Bernard then claims to know the date he is saying that the day occurs only once among the 3 remaining choices, thus telling everyone that Charlie’s birthday is Apr 16, 2000.

Watch this next on Business Insider: A sleep doctor reveals the best time to take a nap

Read more: http://www.iflscience.com/editors-blog/see-if-you-can-solve-these-6-fiendish-brain-teasers-written-by-nsa-employees/

‘Game Of Thrones’ Is All About This Character, According To Math

Who is the real star of “Game of Thrones?”

Set across the sprawling Seven Kingdoms and beyond, it can be tricky to pinpoint the books’ and subsequent hit TV show’s most important character.

But mathematicians from Macalester College in Saint Paul, Minnesota, think they’ve figured it out. And the answer is…

Yes. Tyrion Lannister — portrayed in the HBO show by Peter Dinklage — is reportedly the glue binding the entire fantasy epic together.

Macalester College’s associate professor of mathematics Andrew J. Beveridge and undergraduate student Jie Shan applied “network science” to the third installment (A Storm of Swords) of author George R.R. Martin’s epic “A Song Of Ice and Fire” novel series.

“We opted for this volume because the main narrative has matured, with the characters scattered geographically and enmeshed in their own social circles,” they wrote in their paper, published on the Mathematical Association of America’s website last week.

They analyzed the interconnection between all the characters, and linked them together every time they appeared within 15 words of one another. This diagram is the end result:

Tyrion appears to be the principal character. Jon Snow, portrayed by Kit Harington in the TV show, is close behind.

Perhaps surprisingly, Sansa Stark (played by Sophie Turner) is also in the running. “Other players are aware of her value as a Stark heir and they repeatedly use her as a pawn in their plays for power. If she can develop her cunning, then she can capitalize on her network importance to dramatic effect,” the researchers wrote.

Daenerys Targaryen, played by Emilia Clarke, was also deemed important — but not so much as the others, due to the way her character appears to be isolated.

“Game of Thrones” Season 6 premieres Sunday, April 24, on HBO.

Read more: http://www.huffingtonpost.com/2016/04/03/game-of-thrones-math-real-star_n_9603534.html

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The Maths Behind Impossible” Never-Repeating Patterns

The Conversation

Remember the graph paper you used at school, the kind thats covered with tiny squares? Its the perfect illustration of what mathematicians call a periodic tiling of space, with shapes covering an entire area with no overlap or gap. If we moved the whole pattern by the length of a tile (translated it) or rotated it by 90 degrees, we will get the same pattern. Thats because in this case, the whole tiling has the same symmetry as a single tile. But imagine tiling a bathroom with pentagons instead of squares its impossible, because the pentagons wont fit together without leaving gaps or overlapping one another.

Patterns (made up of tiles) and crystals (made up of atoms or molecules) are typically periodic like a sheet of graph paper and have related symmetries. Among all possible arrangements, these regular arrangements are preferred in nature because they are associated with the least amount of energy required to assemble them. In fact weve only known that non-periodic tiling, which creates never-repeating patterns, can exist in crystals for a couple of decades. Now my colleagues and I have made a model that can help understand how this is expressed.

In the 1970s, physicist Roger Penrose discovered that it was possible to make a pattern from two different shapes with the angles and sides of a pentagon. This looks the same when rotated through 72-degree angles, meaning that if you turn it 360 degrees full circle, it looks the same from five different angles. We see that many small patches of patterns are repeated many times in this pattern. For example in the graphic below, the five-pointed orange star is repeated over and over again. But in each case these stars are surrounded by different shapes, which implies that the whole pattern never repeats in any direction. Therefore this graphic is an example of a pattern that has rotational symmetry but no translational symmetry.

image-20160811-11006-1xfss0s.gifPenrose tiling.PrzemekMajewski,CC BY-SA

Things get more complicated in three dimensions. In the 1980s, Dan Schechtman observed an aluminium-manganese alloy with a non-periodic pattern in all directions that still had rotational symmetry when rotated by the same 72-degree angle. Until then, crystals that had no translational symmetry but possessed rotational symmetry were in fact inconceivable and many scientists did not believe this result. In fact, this was one of those rare occasions when the definition of what is a crystal had to be altered because of a new discovery. In accordance, these crystals are now called quasicrystals.

Irrational number

The never-repeating pattern of a quasicrystal arises from the irrational number at the heart of its construction. In a regular pentagon, the ratio of the side length of the five-pointed star you can inscribe on the inside of a pentagon, to the side of the actual pentagon is the famous irrational number phi (not to be confused with pi), which is about 1.618. This number is also known as the golden ratio (and it also satisfies the relation phi = 1+1/phi). Consequently, when a quasicrystal is constructed with tiles that are derived from a pentagon like the ones Penrose used we observe rotational symmetry at 72-degree angles.

image-20160812-16364-11u0p6u.png

Quasicrystal lattice structure. Author provided

We see this five-fold symmetry both in the image of the quasicrystal as the ten radial lines around the central red dot (above), and also in the scale model of the central part of the quasicrystal made with Zometool (below). In the model, it helps to think of the white balls to be the locations where we would find the particles/atoms of the crystal structure and the red and yellow rods to indicate bonds between particles, that represent the shapes and symmetries of the structure.

In our recent publication, we identified the two traits that a system must have in order to form a 3D quasicrystal. The first is that patterns at two different sizes (length-scale) which are at an appropriate irrational ratio (like phi) both occur in the system. And second that these can influence each other strongly. In addition to the never-repeating quasicrystal patterns, this model can also form other observed regular crystal structures such as hexagons, body-centered cubes and so on. Such a model makes it possible to explore the competition between all these different patterns and to identify the conditions under which quasicrystals will be formed in nature.

image-20160811-11006-xay0sy.jpg

The structure of a quasicrystal. Author provided

The mathematics behind how such never-repeating patterns are created is very useful in understanding how they are formed and even in designing them with specific properties. That is why we at the University of Leeds, along with colleagues at other institutions, are fascinated with research into such questions.

However, this research isnt just a conceptual mathematical idea (although the mathematics behind it is addictive) it has great promise for many practical applications, including making very efficient quasicrystal lasers. This is because, when periodic crystal patterns are used in a laser, a low-power laser beam is created by the symmetry of the repeating pattern. Having defects in the crystal pattern or alternatively using a never-repeating quasicrystal pattern at the output end of a laser, makes it possible to create an efficient laser beam with high peak output power. In other applications, some researchers are even considering the reflective finishes that quasicrystals might create if added to household paint.The Conversation

Priya Subramanian, Research Fellow Applied Mathematics, University of Leeds

This article was originally published on The Conversation. Read the original article.

Read more: http://www.iflscience.com/editors-blog/the-maths-behind-impossible-never-repeating-patterns/

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Can you solve it? Are you a puzzle Olympian?

Get those neurons out the blocks

Hello guzzlers,

Today, two puzzles inspired by the Olympics:

1) The 100m final

Early this morning Usain Bolt won gold in the 100m.

Lets assume that Bolt won the race by 10m from the second placed runner, Justin Gatlin. And now lets run the final again, but this time Bolt will start 10m behind the start line. If both Bolt and Gatlin run at the same constant speed in the second race as they did in the first, who wins?

2) The mystery games

(For the purposes of this puzzle, any resemblance to real persons, living or dead, and Olympic sports, actual or obsolete, is purely coincidental.)

Nafissatou, Jessica, and Brianne are taking part in an athletics competition that involves at least two events. In each event the winner gets G points, second placed gets S points and third place gets B points, where G, S and B are whole numbers, nonzero, and G> S>B. No event is tied. Nafissatou scores 22 points in total. Jessica and Brianne score 9 points each in total. Jessica wins the 100m hurdles. Who is second in the javelin? How many events are there?

Ill open comments at noon, and post the answers at 5pm BST. Please dont post the answer in the comments before 5pm because this spoils it for the many people who want to work out the answers themselves. Thanks!

I post a puzzle here on a Monday every two weeks. If you want to propose a puzzle for this column, please email me Id love to hear it.

Im the author of three popular maths books including Alexs Adventures in Numberland. My new book Football School: Where Football Explains the World is for children and out on Sept 1. You can check me out on Twitter, Facebook, Google+, my personal website or my Guardian maths blog.

Read more: https://www.theguardian.com/science/2016/aug/15/can-you-solve-it-are-you-a-puzzle-olympian

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The 12 Most Mind-Melting Artworks Headed To Burning Man This Year

Approximately 525 years ago, an Italian polymath named Leonardo da Vinci completed a pen and ink drawing called “The Vitruvian Man.” The piece depicts a nude man in two superimposed positions, inscribed in a circle and a square. Inspired by the work of an architect named Vitruvius, Leonardo created the piece as a manifestation of the ideal human body, the principal archetype of proportionality. 

Fast forward to 2016, when Leonardo’s iconic contribution to the fields of art, science, mathematics and the place where the three intersect will be honored in the most dream-like of ways. In the middle of Nevada’s Black Rock City, approximately 70,000 people will join together to burn the Man to the ground. 

That’s right, this year’s Burning Man theme is Leonardo da Vinci’s Workshop, paying homage to the artist and his crew who transformed 15th- and 16th-century Florence through ingenuity, creativity and crazy skill.

Over 300 artists have registered to bring their most surreal, hallucinatory and utopian Leondardo-inspired visions to life on the large-scale stage that is the playa. There are giant butterfly sculptures. There are floating LED dancing women. There are most certainly breathing robot alligator puppets.

We’ve rounded up 12 of the projects most likely to make Leonardo say, “Whoa, what?” Each artist or artist collective contributed a brief description of their project, along with a quote describing how the piece taps into the overall spirit of Burning Man. Take a look below. 

1. FLOCONS

by Anna-Gaelle Lucy Marshall and Dusty Visions (Portland, Oregon)

Photo by AG Lucy Marshall

Project description: “’FLOCONS’ is a dynamic and evolving series of large-scale, ornament-like sculptures, imagined and built by a collective of artists to be displayed in a garden-like manner. [It] is a shared creative experience in participatory art that displays the beauty of human collaboration and communal effort. As participants approach the series, they will find a collection of large free-floating sculptures, each as unique as the artists who created them.”

Why it captures the spirit of Burning Man: “We believe in the positive impact of community-driven participatory art projects and seek to support the creation of pieces that can involve a wide variety of individuals and groups from concept to completion. ‘FLOCONS’ was designed as a dynamic and evolving series; the metal hangers providing a blank canvas for endless creative expression in a unique format that is transportation-friendly, allowing for multiple displays year-round.”

Learn more about “FLOCONS” here and support the project on IndieGogo here.

2. Firmament 

by Christopher Schardt (Oakland, California)

Photo by Christopher Schardt

Project description: “A vast, overhead canopy of LEDs displays celestial, playful, psychedelic, majestic images while classical music emotionally supports an enveloping, comforting, communal environment below.”

Why it captures the spirit of Burning Man: “’Firmament’ captures an aspect of Burning Man that is increasingly hard to find. It is an oasis of calm in a sea of thumping art cars, a place of beauty rather than spectacle. Thousands of people lie under it every night, enjoying the flowing visuals and peaceful music as long as they like, before returning to the cacophony that is Burning Man at night.”

Learn more about “Firmament” here. 

3. Mechan 9

by Tyler Fuqua (Eagle Creek, Oregon)

Photo by Jason Hutchinson

Project description: “’Mechan 9’ is a giant fallen robot that is half-buried in the ground. How he got there, and why he fell in the desert, and most importantly, who built him, can all be revealed if you decode the strange text that is on his body using a special decoder. (A decoder sticker found in a hatch on his head.) The clues will take you on a treasure hunt all over the city where you will learn the history of ‘Mechan 9’ along the way. The robot is over 30 feet long and can be climbed upon. In fact, you HAVE to climb on him to see the secret of his heart chamber!”

Why it captures the spirit of Burning Man: “Burning Man is about discovery. Discovery of one’s self, discovery of new people, and discovery of amazing art. ‘Mechan 9’ captures that spirit by inviting people to decode a strange text and discover the mystery of the giant robot. Who knows what else you will discover on your adventure to unravel the secrets of ‘Mechan 9’!”

Learn more about “Mechan 9” here.  

4. Electric Renaissance (A Tribute to Cadillac Ranch)

by Heliotropics (Santa Cruz, California, and Aachen, Germany)

Electric Renaissance
Rendering: Project. “Electric Renaissance,” Burning Man 2016

Project description: “’Electric Renaissance’ celebrates the advent of the zero-emissions, all-electric vehicles of the future with a salute to Cadillac Ranch, Ant Farm’s iconic installation, the tombstone of the gas-guzzling, gross, polluting dinosaurs of the industrial past. [It was] conceived of by the ‘Electric Renaissance’ team of Jason Guy, Bodo Jülicher, Paulina Kuper, Keith Muscutt, Bob Oberg, Steve Ornellas, Sean Pace, Richard Peifer, Pawel Pietkun, Joslyn Schott, Jacob Stelzriede, Jakub Sztur and Andrzej Sztur.”

Why it captures the spirit of Burning Man: “’Electric Renaissance’ envisions that the Cadillacs, after a harrowing and transformative journey through the bowels of the earth, are being resurrected as zero-emissions vehicles at the dry lake bed ‘playa’ at Burning Man. In keeping with the Leonardo da Vinci theme of the Burning Man Festival this year, ‘Electric Renaissance’ also celebrates human ingenuity in developing technological solutions to societal and environmental challenges. While the original concept of the installation was to use actual electric-vehicle bodies, buried trunk down in the playa, that proved both financially and technically very difficult to negotiate (due in part to the fragility of the playa). So, like most art at Burning Man, ‘Electric Renaissance’ will be made of flammable materials and destined to end in a blaze of glory (unless someone spares it that fate).”

Check out “Electric Renaissance” on Instagram. 

5. Dust City Diner

by David Cole and Michael Brown (Oakland, California)

Vertumnus
Servin’ up sandwiches and sass, you’ll find us when you need us most.

Project description: “The ‘Dust City Diner’ is a magical 1940s-style diner, complete with a quilted stainless steel backsplash, red leather stools, a gurgling coffee maker, the Ink Spots and the Andrews Sisters on the radio, and of course blue plate specials. Created for Burning Man 2008’s theme “The American Dream,” the ‘Dust City Diner’ serves hot coffee and grilled cheese sandwiches from dusk till dawn at an unannounced location in the deep playa. Our goal is to provide a special respite for those who want to sit, read the paper and enjoy the company of other weary travelers.”

Why it captures the spirit of Burning Man: “We love the idea of a gift economy, and what would be more appreciated than a classic grilled cheese sandwich and a piping cup of strong coffee in the middle of the night? The idea began as we reflected upon the strange experience of being surrounded by people and activity at Burning Man, but still feeling somewhat isolated and not connecting with those around us. The project was to create an intimate oasis for an evolving cast of strangers who, for as long as they occupy a chair, are the ‘regulars’ who give character to all diners across the expanse of America. The special appeal of building something that one just stumbles upon and perhaps never finds again is one of the things that is most particular to Burning Man.”

Learn more about the “Dust City Diner” here. 

6. Heart of Gold

by HYBYCOZO (Yelena Filipchuk & Serge Beaulieu) (San Francisco, California)

Photo via Yelena Filipchuk

Project description: “A series of geometric sculptures based on Leonardo da Vinci’s drawings that play with your sense of perspective and warp your perception of the inner and outer dimensions of the self.”

Why it captures the spirit of Burning Man: “Our work seeks to capture the spirit of wonder, larger-than-life installation, and the curiosity of geometric, out-of-the-box thinking. Building and creativity are the lifeblood of the Burning Man community and we hope to make honest artwork that invites people to think about construction and design.” 

Learn more about HYBYCOZO here.

7. Horizon Lines

by Tyler Buckheim (Key West, Florida)

By Tyler Buckheim

Project description: “At first look ‘Horizon Lines’ appears to be a cluster or ‘forest’ of 25 wooden posts rising out of the playa. The top portion of the posts are painted black, with a golden line between the black and raw wood sections, and the sides of each post painted to a different level. When participants come to the properly marked location looking directly north, south, east or west at the sculpture, the painted portions visually align with the horizon or mountain range behind it. The golden stripe creates a gilded horizon line. The moment the paint on each post and the natural backdrop line up, you will see the optical illusion of the merging of the foreground and background to make one whole image.”

Why it captures the spirit of Burning Man: “Burning Man thrives on individual perspective and experiences. Each person chooses to be present and fully immersed in this bizarre and wonderful culture. The sculpture reflects this essential part of the burning man experience by interacting with the viewer. Making each person a necessary element of the sculpture and allowing them to experience the morphing of their surroundings based on their own movement. Radical self-reliance is one of the key principles of Burning Man and, as a part of that idea, the sculpture itself relies on each individual’s involvement and interaction.”

8. Dreams of Flight 

by Michael Gard (San Francisco, California)

by Michael Gard
One of the figures he’ll be flying.

Project description: “My project is wire figures created in aluminum with internal LED lights. These will be flown around the playa at night under large black helium balloons.”

Why it captures the spirit of Burning Man: “One of the finest of many spirits Burning Man represents to me is wonder. This project is pretty mysterious from even 50 feet. The night sky really hides the balloons, giving the impression of mysterious floating, dancing forms. Coming closer, people figure it out, but then dig the ingenuity. It’s fun to interact with people after being approached with that level of novelty.”

See a video of “Dreams of Flight” at Burning Man 2014 here. 

9. The MechaGator

by Ryan S. Ballard (New Orleans, Louisiana)

by Ryan Ballard

Project description: “MechaGator” is a 12-foot-tall fire-breathing robot alligator puppet controlled with levers and foot pedals that you operate by mounting her little pink pony saddle on her tail. She has disco balls, little pink wings and a unicorn horn. And she has a serious appetite for propane.”

Why it captures the spirit of Burning Man: “’MechaGator’ captures the spirit of Burning Man in her flaming interactive whimsy. She brings joy to everyone who encounters her.”

10. Imago

by Kirsten Berg (Berkeley, California)

By Kirsten Berg

Project description: “Blue-mirror-steel butterflies hover 17 feet over the desert, wings lifted as if just alighted, yet ready for takeoff. Stepping beneath the delicately-perforated, arching wingspan, vivid blue light scatters, encircling us in a vaulted, luminous space of geometric patterns/reflections. ‘Imago’ holds a space, in structure and feeling, that is light and uplifting, like an intimate temple or futuristic shrine. [The] design was specifically inspired by the arching, peaked roof shapes of Buddhist temples and ‘spirit houses’ found in Thailand, where I live part of the year, and the blue iridescence of Morpho butterflies. With ‘Imago,’ I wanted to bring the static, geometric forms of the temples together with the organic living form of the butterfly, all to create a space to immerse and reflect in light.”

Why it captures the spirit of Burning Man: “The elevated butterflies are easily resonant metaphors for the transformative experiences that compel so many migrations to Burning Man: immersion into a place of imagination, to be renewed, transformed. Butterfly metamorphosis mirrors our playa experiences: intense preparations, followed by the suspension of mundane life, to immerse into a dreamlike space that is safe yet undefined, full of potential. When the time of cocooning is over, we emerge, high on imagination. Just as the mature butterfly, known as an imago, returns to the wider world, ripe for creation and pollination.”

Learn more about “Imago” here. 

11. Piazza di Ferro

by Iron Monkey Arts (Seattle, Washington)

Rendering by Tabasco Mills

Project description: “This year the Iron Monkeys will pay homage to our craft by building a functional, participatory blacksmith shop on the playa. Encircling the shop will be a public gathering space, encouraging the community to observe, discuss and contribute to what is happening in the shop. Inspired by the theme, the Iron Monkeys decided that this is the year to share our craft at Burning Man. 2016 also marks the 10th anniversary of the Iron Monkeys, and in celebration, we want to encourage an important element of our philosophy: anyone can blacksmith. The theme for the event this year is a great opportunity to share our craft and give participants a chance to experience blacksmithing.” 

Why it captures the spirit of Burning Man: “Our piece captures the spirit of Burning Man by imbuing participants with the belief that ‘I too can do.’ We believe anyone can blacksmith, and will spend all week teaching this old-world craft to those who enter the shop. This is interactive art at its dirty, sweaty core.”

Learn more about the “Piazza di Ferro” here. 

12. Renaissance  

By Pink Intruder (Miguel Arraiz and David Moreno; Valencia, Spain)

Pink Intruder
Renaissance model

Project description: “’Renaixement (Renaissance); will be an art installation at Burning Man 2016 that brings, for the first time, the aesthetic of Las Fallas to Black Rock City. Funded in part by an Honorarium Grant from Burning Man, this “falla on the playa” will use a contemporary language in its external appearance but also express the essence of Valencian culture and the Fallas tradition. With this year’s Burning Man theme and [Leonardo’s] connection with the Borgia family and Valencia’s Silk Exchange Market building, a contemporary architectonic cardboard pavilion will be built on the playa. A cardboard tube structure of over 50,000 pieces, the mosaic ground floor mixed with traditional Fallas Festival cardboard sculptures [will show] the traditional way of working in our city, Valencia, Spain.

Why it captures the spirit of Burning Man: “Through all the year, thousands of people work to make possible the Fallas festival. At this point, the project meets the spirit of the Burning Man: the community. The base of the ‘Renaissance’ pavilion the ground floor, with thousands of mosaic pieces has been assembled by kids and social [organizations in Spain].”

Learn more about “Renaissance” here. 

Read more: http://www.huffingtonpost.com/entry/burning-man-art-2016_us_57b1c498e4b069e7e505ca4f?section=&

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Trypophobia: The Fear Of Holes Driven By The Internet And Mathematics

Since the advent of the internet, people have been able to discuss their symptoms with others globally. Sometimes people with very unusual symptoms discover others with similar experiences, which they are then able to discuss without fear of ridicule. Discussion forums and support groups are formed and eventually a new medical condition may be recognised. A case in point is visual snow, which individuals experience as bright dots persistently floating like snow across their vision. Another is trypophobia.

Trypophobia a fear of holes is a condition which triggers individuals to suffer an emotional reaction when viewing seemingly innocuous images of clusters of objects, usually holes. The condition was first described on the internet in 2005 though it is not yet a recognised medical diagnosis.

The images responsible for the emotion include natural objects such as honeycomb or the lotus seed head, and man-made objects such as aerated chocolate or stacked industrial pipes viewed end-on. Despite their seemingly innocuous nature, images such as these (ideal for sharing on the internet) can induce a variety of symptoms including cognitive changes that reflect anxiety, bodily symptoms that are skin-related (such as itchiness and goose-bumps), and physiological changes (such as nausea, a racing heart, or trouble catching breath).

The images that induce the emotional reaction would not normally be conceived of as threatening so, in this respect, trypophobia differs from many other phobias.

Mathematical properties

Phobias are anxiety disorders that are normally thought to arise because of learning (a dog bite may lead to a fear of dogs) or because of innate evolutionary mechanisms such as may underlie a fear of spiders and snakes. Usually, there is a threat, specific or general, real or imagined.

In the case of trypophobia, there is no obvious threat, and the range of images that induce the phobia have very little in common with one another, other than their configuration.

It appears that it is this configuration that holds the key to the emotion that the images induce. Individuals who do not profess trypophobia still find trypophobic images aversive, although they do not experience the emotion. They do so because the configuration gives the image mathematical properties that are shared by most images that cause visual discomfort, eyestrain or headache.

Images with these mathematical properties cannot be processed efficiently by the brain and therefore require more brain oxygenation. In a paper, Paul Hibbard and I proposed that the discomfort occurs precisely because people avoid looking at the images because they require excessive brain oxygenation. (The brain uses about 20% of the bodys energy, and its energy usage needs to be kept to a minimum.)

Uncomfortable? You may have a dose of trypophobia. Theen Moy, CC BY-NC-SA

So trypophobic images are among those that are intrinsically uncomfortable to look at, and we are now investigating why it is that some people and not others experience an emotional response.

Images of contaminants such as mould and skin diseases can provoke disgust in most people, not just those with trypophobia. The disgust is probably an evolutionary mechanism that promotes avoidance and has survival value.

Images of mould and skin lesions have mathematical properties similar to those of images that are trypophobic and our current work explores whether they also induce a large oxygenation of the brain in addition to being generally uncomfortable. Perhaps discomfort is a useful mechanism not only for avoiding excessive oxygenation, but also for rapidly avoiding objects that provide a threat in terms of contamination. It may be that in people with trypophobia, the mechanism is overworking.

Top image credit:Feeling nauseous? Leo Reynolds/Flickr, CC BY-NC-SA

Read more: http://www.iflscience.com/brain/trypophobia-fear-holes-driven-internet-and-mathematics

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