Der Beitrag Few tile attempts erschien zuerst auf Heidelberg Laureate Forum.
]]>Tilings, also known in maths as tessellations, come in many different types. As well as tilings made entirely using the same repeated shape (called a regular tiling if the shape in question is a regular polygon, like a square) you can also have semiregular, or uniform tilings – with two or more different regular polygons, meeting in different ways to create a repeating pattern.
Sadly if you restrict yourself to regular tilings, your options (for a 2D wall at least) are quite limited – you can only have squares, hexagons or equilateral triangles. None of the other regular polygons are capable of tiling the 2-dimensional plane, and if you want more variety you’re forced to mix and match your shapes.
Uniform tilings have an additional property that every vertex – corner where tiles meet – is the same. Even with two or more shapes, it’s possible to have this condition, and there are several ways to combine equilateral triangles, squares and hexagons, and even dodecagons, to form such a tiling.
Such tilings are sometimes also called Archimedean tilings, by analogy with Archimedean solids: ones made up of different types of shape, but with each vertex looking the same, like the cuboctahedron pictured here.
Other uniform tilings, called k-uniform, have k different types of vertex, and there are many hundreds of different examples of such tilings. They’re always made using two or more different shapes, with the sizes chosen so that the lengths of the edges all match up to form tilings.
The exception to this is if you relax the requirement that tiles meet at corners – if you’re not bothered about this, you can make non-edge-to-edge tilings, like the Pythagorean tiling. It’s made from two different sizes of square, and is so-called because the relative sizes of the two squares match the lengths of two sides of a right-angled triangle. The tiling itself contains a proof of Pythagoras’ theorem: if the two squares have side lengths A and B, the two shorter edges of the triangle, C is the hypotenuse, and you can see that A² + B² = C² by rearranging the pieces into whole squares.
You can’t tile a plane with regular pentagons, but you can if you’re prepared to use non-regular ones, and as recently as 2015 a new way of tiling the plane using similar pentagons was discovered, bringing the total number of ways known up to 15.
And if you don’t like your tilings to have a repeated pattern, you can always use an aperiodic tiling – one which is made up of the same shape or set of shapes, but doesn’t have a repeating unit and can go on forever without ever repeating the same way. The most famous example is the Penrose Tiling, named after mathematician Roger Penrose, and it consists of two shapes which can fit together but can also tile the whole infinite plane without making a repeating pattern.
If you allow your square tiles to be non-regular, you can have rectangular tiles, and if they happen to be twice as wide as they are tall, you can make domino tilings. These are ways to arrange tiles measuring 2 by 1 units, and for an n × m rectangle you need (n × m)/2 tiles.
Not all shapes can be tiled in this way – a famous puzzle considers a chessboard, which has been ‘mutilated’, so that the top left and bottom right corners have been cut off. It’s asked if it’s possible to tile such a chessboard with dominoes. I won’t tell you the answer, but you might want to consider what colour the two squares that have been cut off were, and what colour squares a domino on a chessboard will cover up…
So some shapes can’t be tiled, but others can – and any rectangle whose sides aren’t both odd can be covered (if both sides have an odd length, the total number of squares will be odd, so it’s not possible with a whole number of dominoes). But if it is possible, how many ways can it be done?
To start with, consider a 2 × n rectangle, and count how many ways you can arrange tiles to fill it. For n=0, there’s only one way to fill the rectangle (you don’t). For a 2 × 1 rectangle, you just need one domino, and there’s only one way to put it in; for 2 × 2, you can arrange the two dominoes vertically or horizontally. If you carry on with this, you might see an interesting pattern start to form in the number of possible tilings…
My bathroom floor is sadly more than 2 tiles wide, so I’ll need to consider the general case – luckily, it was proven twice, by two separate mathematicians in 1961, that for such an (n × m) rectangle there aredifferent ways to arrange the tiles. If I’m going to narrow it down to one, I’ll need to put some constraints on.
In Japan, tatami are rectangular floor tiles in this 2 by 1 shape, which are often used to create floors by arranging them in a tiled pattern. Interestingly, it’s considered bad luck (‘inauspicious’) to place the tatami so that four meet at a corner, and they must therefore be arranged so that only three mats meet at any given corner. For rooms with odd sides, half-tatami can be used in the corner, and the sizes of rooms are sometimes given in terms of how many tatami they are tiled with – in parts of Japan, shops are traditionally 5½ tatami along the edge, and tea rooms are 4½.
Another condition you can place on domino tilings is that they can be ‘fault-free’ – a fault is a line all the way across the shape which no tiles cross, so that in theory the tiling could be split into two pieces along this line, like a tectonic fault line. In 1981, Ron Graham worked out the conditions required for an n × m rectangle to have a fault-free tiling by dominoes – an m × n rectangle of area greater than 2 has a fault-free tiling if and only if (m × n) is even, m and n are both 5 or bigger, and m and n do not both equal 6.
Specifically, it’s not possible to tile a 6 × 6 square using 36 ÷ 2 = 18 dominoes. A proof of this goes as follows: if you want no faults, each of the 10 lines running across or down the middle of the grid between cells must have dominoes that straddle it. Since the square has 6×6 cells, any such line will divide it into two halves containing an even number of cells, which means there must be an even number of dominoes cut in half by each line, otherwise you’d end up with an odd number of squares on one side. Zero is even, but if any of these lines cuts 0 dominoes in half, it’d be a fault – so each of the 10 lines has to cut at least two dominoes. Therefore you need the tiling to contain at least 20 dominoes. This can’t happen, as the square isn’t big enough! QED.
Unfortunately for me, as exciting as all these tilings are, not many of them will really work for our bathroom floor. I can probably find square tiles in the tile shop, probably some rectangles and if I’m really lucky maybe some hexagons. But all these other tilings are probably not possible using simple tiling techniques (and our decorator gets a bit angry if you start to suggest the idea of cutting unconventional shapes out of regular tiles, or insisting you lay the tiles in strange layouts). So we needed to come up with something that would work without requiring funky shaped tiles or layouts, but that still has an interesting mathematical twist.
One option we can use to make it more interesting is to colour the tiles – they sell square tiles in a range of colours, so we could use them to create some kind of interesting pattern in colour rather than shape. Our decorator is happy with normal square tiles laid according to a particular colour layout, so we just need to give him a piece of paper with a pattern to follow.
As we already have previous experience of this (see: my Aperiodical post from 2015), we decided to use black and white square tiles to spell out a message in binary, using five tiles for a letter of the alphabet given by a number between 1 and 26.
I won’t give away the message we’ve encoded in the bathroom, but if you keep an eye on my Twitter feed I might post some photos of it once it’s finished!
Der Beitrag Few tile attempts erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag Cake and Coincidence erschien zuerst auf Heidelberg Laureate Forum.
]]>Last week I was performing at the Cheltenham Science Festival in the UK, where I co-presented a new family show about probability and games. One of the games we played was, “How many people do we need to ask ‘when is your birthday?’ (working backwards from the front of the audience) until we find two people with the same birthday?”
This is a problem you can analyse mathematically, making some sensible assumptions. First, you can simplify messy reality by assuming that there are exactly 365 days in a year (rather than the slightly less helpful but more realistic 365.2425) – meaning, we ignore the concept of leap years. This is mildly unfair on people who were born on the 29th of February, but their chances of winning our game are much lower anyway, so they’ve probably already given up.
The second simplification we can make is that everyone’s birthday has an equal or roughly equal probability of being on any one day of the year. This seems to be broadly a sensible assumption – but in reality there are some practicalities which can skew this distribution.
For instance, many hospitals have more facilities open during the week than at the weekend, including those for induced or Caesarian births, meaning on average more birthdays tend to be on a weekday than at a weekend – but this will average out across years, as the weekdays fall on different days. It’s also known that statistically the most common time of year for birthdays is around September, falling roughly nine months after the dark, stay-indoors winter period when people presumably… umm… stay indoors more.
But everyone is subject to the same probability distribution, and factoring it in would make a much more complicated model, and not much difference to the results. So let’s assume the probability is equal across the year, to simplify our calculations. This means everyone has a 1/365 chance of their birthday being on any particular day, and for any given person you might meet, there’s a 1/365 chance they have the same birthday as you.
What’s interesting about the problem as applied to our game is, the chances of a birthday coincidence happening, and happening before we get very far through the room are much higher than many people expect them to be. It’s become a well-known ‘surprising fact’ among mathematicians, that the number of people you need before your chances of a birthday coincidence go above 50% is a mere 23 people.
When asked to guess how many people they thought we’d need, many in the audience picked numbers over 50, and in some cases over 100. But 23 people is actually enough for a coincidence to become more likely than not.
Mathematically, an easy way to calculate the probability is to start with one person and add other people one at a time, each time checking the probability of a birthday coincidence. In fact, it’s even easier to calculate the probability of no coincidence occurring, and then invert this – trying to include all the possible combinations of two, three and more people having the same birthday makes the calculation become complicated quite quickly.
This can be continued to obtain the probabilities for each number of people; these can then be converted to percentages and subtracted from 100% to get the probability of there being a coincidence.
Number of people | Calculation | Probability of no coincidence | Probability of coincidence |
2 | 364/365 | 99.73% | 0.27% |
3 | 364/365 × 363/365 | 99.18% | 0.82% |
4 | 364/365 × 363/365 × 362/365 | 98.36% | 1.64% |
… | … | … | … |
21 | 364/365 × 363/365 × … × 345/365 | 55.63% | 44.37% |
22 | 364/365 × 363/365 × … × 344/365 | 52.43% | 47.57% |
23 | 364/365 × 363/365 × … × 343/365 | 49.27% | 50.73% |
… | … | … | … |
30 | 364/365 × 363/365 × … × 336/365 | 29.37% | 70.63% |
If you look down this table, you’ll find that the probability tips over 50% at just over 23 people, and for 30 people it’s a little over 70%. This is impressively high, and for many it’s hard to believe – for 60 people in a room, the probability is around 99.6%, and you can probably do quite well making bets with people that there are two people with the same birthday in rooms of 30-40 people, where your chances are actually pretty high.
One of the main reasons why this probability surprises people is that they’re asking themselves the wrong question. When you ask them to predict the chances of a birthday coincidence in a room, some part of them imagines themselves going into a room and finding someone with the same birthday. But this is quite an unlikely event – you’re looking for people with a particular birthday, which is 1/365 × the number of people in the room. This means you’d need 365/2 = 182.5 people or more for this to be more likely than not.
But in the question we’re actually asking, the difference is subtle. I want to know the chances of any two people in the room having the same birthday – and I don’t care what birthday it is. So every pair of people in the room is a possible hit – and the number of possible pairs in a room of people grows much more rapidly than the number of people. For three people, there are 3 pairs, but for 4 people there are 6, and for 5 people there are 10 possible pairs.
With 23 people in the room there are (23×22)/2 pairs, because there are 23 choices for the first person, and 22 for the second, but this counts each pair twice as they could occur either way round. This gives 253 possible pairs of people, and with some wiggle room to take into account the coincidences involving more than two people, this is enough to put the overall chance of any kind of coincidence as over half.
In fact, as eagle-eyed maths/football fan Mitchell Stirling pointed out on Twitter earlier this month, following the announcement of the World Cup 2018 squads: of the 32 different 23-player squads put forward by the countries of the world, 15 of them contain a birthday coincidence. Impressively close to 50%!
In our audience at Cheltenham, however, something even more impressive happened. When we reached around half-way along the second row, a child stated his birthday – 12th February – and a woman seated next to him – who turned out to be his mother – also gave her birthday as 12th February! This was sufficient to win our prize, which was a bar of chocolate each.
But then, as if this coincidence wasn’t impressive enough, the pair informed me that the father, who wasn’t there on the day, also has the same birthday! This means a mother and father, and their son, all share a birthday. What are the chances? (That’s a dangerous question to ask a mathematician, because they will just tell you the answer).
For three randomly selected people, the chances of them all having the same birthday are (1/365×1/365) = 1/133,225. The first person’s birthday, whatever it might be, has a 1/365 chance of being shared by each subsequent person, and we multiply these together as they’re independent events.
This is why in news stories about families who share these kinds of remarkable coincidences, huge probabilities are often stated – and these kinds of stories are pretty easy to find, as they make enjoyable news and happen relatively often.
Eastbourne baby born on same date as his two siblings, BBC News, October 2010
Three generations of Canada men share same birthday, BBC News, September 2017
Family have four children with same birthday at odds of more than 133,000 to one, The Telegraph, January 2014
(In this case, two of the four children are twins, so this is the same calculation as for our game-winning family.)
Three generations of Littleton family boys all born on July 1; family celebrates rare coincidence, from ABC Denver 7, July 2017
Here the probability is given as ‘a one in a 33,000 chance’, presumably having misinterpreted the number as 1/33,000 and not one in 133,000.
Some coincidences turn out to be even more impressive:
Can your family beat this incredible birth coincidence?, KidSpot, August 2016
Here a father, two sons and a granddaughter all share the same birthday – despite the odds being one in 48,627,125, nobody thought to mention this in the article.
Some, despite still apparently warranting news coverage, are actually less impressive:
Coincidence? Family welcomes second set of twins born on identical date, The Irish Times, October 2017
There’s actually a 1/365 chance of this happening, if you already have one set of twins, so it’s not really anything to get excited about. The impressive thing here is having two pairs of identical twins in the same family – the chances are around 1/4,489, given that one in every 67 pregnancies on average results in a multiple birth.
In fact, even for the impressive 1-in-133,225, and 1-in-48,627,125 chances of three and four identical birthdates in a family, I’m not really that fussed. In much the same way that a birthday coincidence is much more likely to happen if you don’t care about which actual birthday it is, if you have millions of families around the world having babies, things that happen one time in 133,000 will happen fairly often – and even things that happen one time in 48 million will still happen every once in a while.
It’s also worth considering that our family with mum, dad and child all sharing a birthday might not be as unlikely as you think – depending on how they originally met. I started my probability show with the revelation that my old friend Jimi, who I went to university with, and first met while out in a pub, has the same birthday as me. I also share a birthday with another of my friends from uni, who I briefly dated.
My theory goes as follows: It’s not insignificant that if you have the same birthday as someone, there’s an increased chance you’ll be going out for a drink around the same time, and if you are the kind of person who meets your future spouse in a bar, you’re therefore slightly more likely to marry someone who has the same birthday as you. I imagine this effect is only small – but we’re not necessarily talking about independent probabilities before.
So maybe we want to stick to people whose inclusion, or not, in a family group isn’t influenced by their birthday, because they’re born into it. I looked up the Guinness World Record for most siblings born on the same day, and it turns out it’s five – the Cummins family in the USA has five children all born on 20th February. This happens with probability 1 in 17,748,900,625 (the Guinness website states it as one in 17,797,577,730, presumably because they’ve remembered about leap years and have a more sophisticated model).
Given that 17 billion is more than double the population of the world, and many times the number of families, this is genuinely something that’s actually quite unlikely, and deserving of a prize like an official World Record. I hope they got more than a bar of chocolate each!
Der Beitrag Cake and Coincidence erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag The Bridges of Königsberg erschien zuerst auf Heidelberg Laureate Forum.
]]>People who lived in the city often wondered idly over coffee whether it would be possible to make a journey through the city, crossing all of the bridges, but without crossing any of the bridges twice. Each bridge would be crossed exactly once in some direction – such a feat became known as ‘walking the bridges’. People struggled to find a solution to this problem, and it took a mathematician – Leonhard Euler, who lived in nearby St Petersburg – to find the answer.
Euler was a Swiss mathematician, physicist, astronomer, logician and engineer, born in 1707 in Basel, Switzerland. He was well known for making many great and deep contributions to mathematics, and has more published works than any other mathematician ever. He also introduced much mathematical terminology and notation that is still used today.
Among his many contributions was the theory of graphs, which was inspired by his work on the Bridges of Königsberg problem – the paper he published on the problem in 1736 is regarded as the first in the history of graph theory.
Euler was able to answer the question of the Bridges of Königsberg using his new-found theory – by first working out a general rule for all similar problems, then applying this understanding to the specific seven bridges problem. Often in mathematics, generalising a problem to see how it works in different cases can be a great way to get an understanding of the overall problem, and in this case it spawned an entire branch of mathematics.
Graphs are networks of vertices (points) connected by edges (lines), and can be used to model all kinds of real-world and theoretical problems. The idea of a graph is to show how objects are connected without worrying about actual distances, angles or the geometric shape of the graph. Euler called it the ‘geometry of position’, as position rather than distance had become the significant factor.
You could draw a vertex for each of the numbers 1-10 and connect them with an edge if the numbers share a common factor; you could model the friendships within a group of people using a graph; or you can use them to describe physical networks – graphs are useful in designing, modelling and planning networks such as train lines, utilities supply and delivery routes.
In these kinds of real-world networks, there’s other important information, like the distances between stops, or how long it takes to travel – but you don’t need this information to draw a graph. Many train route maps are drawn using a pure graph, as the actual distances between stations aren’t as important, rather just the connections between the stops.
This difference can be seen in these two copies of the London Tube Map – on the left (image from TFL.gov.uk) the map is laid out to clearly show the connections between the stations, but the positions of the stations are not the same as they are on a real surface map. On the right (image from london-tubemap.com) the positions of the stations and lines are accurate, but this map is less useful to plan a journey – some stations are bunched up close together so it’s harder to make out what’s going on, and there are large gaps between some stations which make the map bigger than it needs to be.
It is possible to store more information in a graph than just the way things are connected – some graphs have arrows connecting the vertices instead of just plain edges, and these are called directed graphs or digraphs – they store information about which way you are allowed to travel along the edge, if this information is important – for example, if the connections you’re modelling are one-way, and a double arrow can be used to indicate a two-way connection. It’s also possible to assign a number (a weight) to each edge, so you could encode distances or travel times, and this is called a weighted graph. The variety of problems that can be modelled using graphs is impressive – and the Bridges of Königsberg problem has the honour of being the first.
The problem of ‘walking the bridges’ in Königsberg can be seen as equivalent to the following problem: if the city is drawn as a graph with a vertex for each part of the city and edges where the bridges connect the different parts, can we find a path which travels along each edge exactly once?
We define an Euler Path as a route which travels along all the edges of a graph, in some direction, exactly once each edge. It may involve visiting a vertex more than once, as long as that vertex has enough edges connected there. (The problem of traversing a graph and visiting all the vertices once is a separate question.) It’s also possible, for some graphs, to create an Euler Circuit – one which travels along all the edges and finishes at the same vertex it started at.
One property of a graph that’s relevant to this is the number of odd and even vertices in the graph – a vertex is odd or even depending on whether it has an odd or even number of edges meeting there. The number of edges meeting at a vertex is called the degree of the vertex.
If a graph has exactly two odd vertices, it’s possible to find an Euler Path starting at one of these two vertices and finishing at the other. If the graph has no odd vertices at all, it’s possible to create an Euler Circuit, starting and finishing at the same vertex (and this can be completed starting anywhere in the graph).
Euler discovered all of this by sketching Konigsberg, and other graph layouts, and comparing their properties to determine exactly what makes a graph possess an Euler path.
So what does this mean for the city of Königsberg? The four islands (or vertices, in the equivalent graph) all have either 3 or 5 bridges meeting there. This means it’s not possible! If you spent a while trying, don’t feel bad – but Euler’s achievement of proving why it’s not possible is in some sense a solution to the original problem. Quite often in maths, if something can’t be done, it’s still satisfying to fully understand why, and in this case it’s lead to the development of some hugely interesting and useful mathematical ideas.
Today, due to historical restructuring after bombing damage in the war and other building since, two of the original seven bridges are no longer standing, and others have been added – such that it is now possible to cross all the bridges of Königsberg (which is now part of Russia, as of 1945, and is called Kaliningrad) in one Euler path. I guess that’s one way to solve a problem – although it has taken hundreds of years!
If you’re dissatisfied with an impossible puzzle, here’s a variation on the original. Imagine the city of Konigsberg is inhabited by two rival families – the Red family, occupying the south bank of the river, and the Blue family, who live in the north. Within the city, each family has their own schloß (castle) marked on the map in their own colour, and there’s also a gasthaus (inn) marked in yellow in the centre.
After seeing these two rivals wreak havoc on the city with their inane bridge building (and quite dismayed at the amount of time people are spending at the gasthaus), the Bishop decides to scupper all their plans by building a tenth bridge. He’d like to make it possible for anyone who lives anywhere in the city to walk the bridges, and return home to their own homes. Where should he build a tenth bridge?
Der Beitrag The Bridges of Königsberg erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag A truly special occasion erschien zuerst auf Heidelberg Laureate Forum.
]]>Imagine a taking a number and subtracting one from it. Then imagine starting with that same number, and finding its reciprocal (calculating the result of 1 divided by the number). Now imagine that if you do each of these things you get the same result. That would be quite a nice coincidence. In fact, there is a number that has this property, and it’s the Golden Ratio.The Golden ratio ϕ is an irrational number, as it contains √5. So it’s interesting that it’s called a ratio, since it’s… not an exact ratio of two things. But if you compare ϕ with 1, the ratio between the two numbers is what’s really Golden. Imagine you have a rectangle which has short side length 1 and long side length ϕ.
If you cut off a 1 by 1 square from this rectangle, the resulting rectangle will be (ϕ – 1) by 1, and given the fact above, this means the smaller rectangle is in the same ratio as the larger one – if you rotate it and scale it up in proportion, you’ll get back the original rectangle. There’s exactly one kind of rectangle for which this works – this one – and it’s called a Golden Rectangle.
Many people claim that this Golden Ratio is somehow a ‘perfect’ ratio – the ‘Divine Proportion’, symbolising perfect beauty and truth, or something. Claims are made about how art, nature and even the human body contain instances of this ratio. Unfortunately, some of these claims are mildly unfounded, and there’s no real evidence they are actually the Golden Ratio (and don’t get me started on Nautilus shells). It seems people just want to find it, and go looking for it in all kinds of places – and their enthusiasm carries them to draw Golden conclusions.
Artists, including Salvador Dali, have been known to employ the Golden Ratio in some famous works – but many others are also purported to have used it, when actually, there’s no evidence they did. If you draw a rectangle on the Mona Lisa enclosing only the subject’s face, you’ll get arrested and thrown out of the Louvre. Nobody seems able to produce proof that the Golden Rectangle is ‘the most visually appealing rectangle’, although it’s a claim that’s often made.
So, if this amazing sturdy fraction doesn’t crop up in nature and art as much as people say it does, why do mathematicians love it so much (and why did we have a celebration to mark ϕ years of marriage)? Read the following round-up of my top 5 places you’ll really actually find the Golden Ratio, and hopefully you’ll begin to understand.
This five-pointed star shape inscribed in a pentagon, sometimes also called a pentangle, has long-standing associations with religious imagery and symbolism across many cultures. But also maths! Pentagons – and by association, shapes made from pentagons, like the dodecahedron – are intricately linked to the Golden Ratio. The four distances marked in the diagram are in the Golden Ratio to each other – that is, a:b, b:c and c:d are all the Golden Ratio. If you knew this already, you win a gold star.
While it’s not hugely well-known, it’s described by Wikipedia as ‘the most common thirty-faced polyhedron’, so if you’re attacked by a 3D shape and it has 30 sides, this is likely the culprit. The rhombuses making up the faces of the shape are called Golden Rhombuses – the ratio of the width to the height of the rhombus is the Golden Ratio, and that’s what makes them exactly the right shape to fit together in this way. If you’d like to build your own, here’s a net.
Believe it or not, you’re almost certainly carrying a Golden Rectangle with you right now. Credit cards, and by extension most wallet-sized cards you might be carrying, are a pretty close Golden Rectangle. You can tell, by taking two cards from your wallet, and giving them to me… no, sorry, placing them on the table with one upright and one horizontal, with their edges touching. Another straight edge will allow you to verify that the line from the bottom left to the top right corner of the horizontal card will then run straight to the top right corner of the vertical card – showing that the small rectangle at the top is in the same ratio as the larger card. Even if you’re not rich enough for a Gold Card, it’s still Golden!
You might be familiar with the sequence of numbers starting with 1, 1 and then continuing to create each term by adding together the two numbers before it – so 1+1=2, then 1+2=3, then 2+3=5 and so on.
This sequence crops up in all kinds of interesting places – in rabbit population modelling, Sanskrit poetry, and in Pascal’s triangle. But if you take successive pairs of entries from this sequence and divide the larger by the smaller, the ratio will be somewhere between 1 and 2, and the further along the list you go, the closer and closer it gets to the Golden Ratio – that’s the limit of the ratios. Even as soon as 89/55, you already get three decimal places of accuracy.
It’s also, as a fun consequence, a handy way to convert between miles and kilometres – since the ratio between 1km and 1 mile is around ϕ, you can use the Fibonacci numbers as a rough conversion chart, by taking the distance in miles and going to the next Fibonacci number for the distance in km. So, 5 miles is around 8km, 13 miles is around 21km and so on.
I hope you’re enjoying this as much as I do.
Der Beitrag A truly special occasion erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag Colouring in like a mathematician erschien zuerst auf Heidelberg Laureate Forum.
]]>One very famous result in the mathematics of colouring problems is the Four Colour Theorem. This states that for any diagram you can draw on a piece of paper, the maximum number of colours you’ll ever need to colour it in, so that any two regions which share an edge are different colours, is four. This means that on a flat surface like a piece of paper, it’s not possible to draw a diagram which needs more than four colours. Go on, try it now. You can’t do it.
This result was proved in the 1970s, and the mathematics underlying the proof is graph theory. Graphs are collections of points joined in a network, with lines between some (but not necessarily all) of the pairs of points, called nodes. In the case of a colouring problem, the shape and size of the regions you’re colouring is unimportant, and the only crucial fact is the way the pieces connect together – which is captured by the structure of the graph, if each node represents one region which needs colouring.
Colouring a graph is as simple as assigning a colour to each node, so that any pair of nodes which are connected use different colours. Colourability (or otherwise) of a graph is often a useful property of the graph to know, and it’s been well studied. If a graph can be coloured with three colours, it’s called three-colourable.
The mathematics that’s been of interest recently, however, doesn’t just involve colouring separate nodes in a network – but every single point of an infinite plane.
The Hadwiger-Nelson problem asks the following question: what’s the minimum number of colours required to colour every single point on the 2-dimensional plane, so that no two points that are exactly one unit of distance apart are the same colour? The answer to this question is called the chromatic number of the plane, and we, um, still don’t actually know exactly what it is.
We’ve known for a while that the maximum number of colours you might need to colour the plane in this way is 7, and we know this because of a proof attributed to American mathematician John Isbell. If you divide the whole plane up into tessellating hexagons, and colour them in using 7 different colours in the pattern shown in the diagram, this is a colouring of the plane which doesn’t have two points distance 1 apart that are the same colour (assuming the hexagons are slightly less than 1 unit across from point to point).
Since 1961, we’ve also known that the minimum this number can be is 4. This was proved using a construction called the Moser Spindle, discovered by mathematical siblings William and Leo Moser. The shape is shown in the diagrams below, and is made up of two pairs of equilateral triangles stuck back-to-back, with an extra line joining the bottom two points. The lengths of the lines in the figure are all 1 unit, which means in order to colour the plane you’d need to be able to colour all the points in this diagram – in particular, the two points at the end of each line need to be different colours. The claim that this can’t be done using three colours is argued as follows.
If the very top point in the diagram is coloured using colour A, then each of the two bottom corners of a triangle attached to it must be coloured B and C, in some combination. This means the very bottom point of that pair of triangles must be coloured A as well. But this argument follows equally for both of the triangle pairs in the diagram, and so both bottom points must be colour A – which is a problem, as they’re 1 unit apart, and therefore can’t be the same colour.
It’s important to note we’re not just dealing with a normal graph here. The Moser Spindle as a graph can’t be three-coloured however you draw it, because of the way it’s connected up – but in this case it’s crucial that the arcs (lines) making up the graph have specific lengths: the Moser Spindle is a unit-distance graph, meaning all the lengths are 1. This fact makes the property of not being three-colourable extend from being true of the graph to being true of the plane as given in the Hadwiger-Nelson problem.
The nice thing about this problem is that it uses ideas from both graph theory and geometry – measuring distance is part of the problem, but graph theory lends its tools to prove things can’t be coloured in this way.
Although the Hadwiger-Nelson problem was first formulated in 1950, all we’ve really managed to prove is that the number is somewhere between 4 and 7. Until now! In the last month, biomedical scientist and, it turns out, mathematician, Aubrey de Grey has announced a new proof, in which he’s constructed a bigger more complex unit-distance graph, increasing the lower bound on this problem from 4 to 5.
The graph de Grey constructed in his paper, which was released on the online preprint server the ArXiV on 8th April 2018, is constructed from smaller, simpler graphs. In the same way that the Moser Spindle is made from two pairs of rigid equilateral triangles, angled so that the distance between their ends is also 1, de Grey’s graph is made up by layering multiple copies of triangulated hexagons. After several layers of repeats, the final graph has 1581 vertices, and provably can’t be four-coloured. So the minimum must be five.
De Grey’s paper also includes discussion of how computer programs can be used to assemble and verify the colourability of the hugely complex graph. Since the release of the paper, the mathematical community has been buzzing with excitement – and proof checking software known as an SAT solver has been used to check the proof.
The next stage is to see if it’s possible to reduce the size of this monstrous graph, to create a neater proof – I expect we won’t quite get to Moser Spindle levels of elegance, but it’s good to try. Several mathematicians, including de Grey himself, have proposed that a good way to do this might be through a Polymath project. Polymath is an online community of mathematicians, who can all collaborate and work on problems together that can easily be split into smaller tasks, and it’s previously been instrumental in this kind of proof optimisation.
The Polymath project on the Hadwiger-Nelson problem, called Polymath16, aims to find smaller non-four-colourable unit-distance graphs, and also to reduce the extent to which the proof depends on computer checking. It’s already discovered a smaller graph which has the same property, with 826 vertices, which can be seen (well, kind of) in the image below.
Work on this project continues, and anyone looking for a minimal way to colour in the plane now has a narrower target to aim at. The mathematics and computer code developed in order to solve this problem enriches the theory of graphs and feeds into other mathematics, but it also takes us a step closer to cracking this gorgeous puzzle.
Der Beitrag Colouring in like a mathematician erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag Abel Prize 2018 awarded to Robert Langlands erschien zuerst auf Heidelberg Laureate Forum.
]]>Langlands was born in 1936 in New Westminster in Canada, and has had a long career in mathematics, working at US universities including Yale, Berkeley and Princeton, where he is now emeritus professor. His early mathematical work included studying automorphic forms – these can be thought of as a generalisation of the idea of a periodic function, one which repeats the same values (such as a sine curve). Langlands studied automorphic forms, and proved various results about them – including some on a particular class of automorphic forms called modular forms. These are a special case where the forms are defined on spaces of 2×2 matrices, and are interesting because while they’re defined and studied as part of the analysis of functions, they’re also connected with number theory.
Number theory is a branch of mathematics which deals with the whole numbers (integers). It includes investigating the properties of numbers, and particular types of numbers, such as prime numbers, or numbers that are made from integers, such as rational numbers (fractions). Number theory is quite often connected to the analysis of certain functions – a famous example is the Riemann Zeta function, which is defined on the complex plane but has deep connections with the distribution of prime numbers.
Langlands’ major contribution to mathematics was made in 1967, when he wrote a letter to French mathematician André Weil describing some of his recent insights on how to connect ideas in number theory to those in automorphic forms – called functoriality. While working at Princeton as an instructor, Langlands met Weil in a corridor on campus, and tried to explain his new ideas. Weil suggested that Langlands put some his thoughts down in writing. The letter was modest but thorough – Langlands expected Weil might readily just throw it in the bin – but it contained 17 pages of mathematical conjectures, generalisations and definitions.
This mathematics formed the basis of what became known as the Langlands Program – a collection of far-reaching and influential mathematical conjectures, relating many areas of maths from number theory to geometry. They have since formed the basis of decades of work by other mathematicians – Fields medals were awarded in 2002 and 2010 for proofs based on Langlands’ conjectures. In fact, the well-known proof of Fermat’s Last Theorem by Andrew Wiles hinged on mathematics laid out in Langlands’ original 17 handwritten pages.
The Langlands program connects automorphic forms to the work of Galois. Évariste Galois (left) was a 19th-century French mathematician, who – in much the same way as Langlands – set down the basis for a whole area of mathematics, again in a single text, which in the case of Galois was written not long before he was killed in a duel. Galois’ work didn’t gain recognition until later, when others realised how much of a breakthrough it was.
Galois theory concerns the symmetry groups of polynomial equations. Given a polynomial – a sum of powers of a single variable, usually x – it’s sometimes possible to find solutions which you can substitute in for x and have the equation balance. Depending on how high the powers of x in the polynomial are, there might be different numbers of solutions possible. For example, a quadratic equation (one containing x²) might have two solutions, and a cubic equation (where the powers go up to x³) up to three solutions.
For quadratics, there’s a well-known formula which can be used to find the solutions – as every school student knows, the quadratic formula for a polynomial ax² + bx + c is given by:In the case of cubic equations, there exists a more complex but similar method to determine the roots. But for certain polynomials, it’s been proven that no such method exists – for example, it’s not possible to solve most quintics (polynomials containing powers up to x⁵) in this way.
For polynomials in general, Galois groups are concerned with the way the polynomial’s solutions are related to each other. For example, consider the following equation:
x² – 4x + 1 = 0
This equation has two solutions: A = 2 + √3, and B = 2 – √3; either of these, substituted in for x, will satisfy the equation. The Galois group of the polynomial describes how these solutions relate to each other, and the ways you can add them together in different combinations – for example, in this case, A + B = 4 and A × B = 1, and in each of these, A and B can be swapped and the equation still holds. This is because A and B are roots which have a kind of symmetry, and the Galois group describes all the symmetries between the solutions.
Galois determined that certain types of Galois groups are possessed by polynomials that can be solved using a formula, like the quadratic equation. Galois called these groups solvable, and Galois theory proves that for all polynomials of degree 4 or lower, the Galois group is always solvable – and hence the polynomial’s roots can be found. For degree 5, some can and some can’t – but those which can are precisely those whose Galois group is solvable. In this way, the group structures are tied to the numerical equations, and Galois theory forms a connection between group theory and number theory. In a similar way, the Langlands program connects the structures of Galois theory to automorphic forms.
Langlands’ work can be seen as a bridge between two seemingly unrelated areas of maths – and increasingly many mathematical discoveries are of this nature. The fact that this kind of work can be so impactful underlines the extent to which mathematics is actually an intricate web of structures and concepts, all of which connect to each other in unexpected ways, and Langlands was a pioneer of this kind of thinking. His work in the 1960s laid the foundation for an entire field of study, and he deserves this recognition for his incredible insights.
Der Beitrag Abel Prize 2018 awarded to Robert Langlands erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag How the HLF helped to pave a path to CERN erschien zuerst auf Heidelberg Laureate Forum.
]]>Pivotal moments in life can be elusive, but when they occur, there is no denying that they will influence the future. They almost never occur in a vacuum, as most often they are links in a chain of events that precipitate them.
Kamil Krynicki, a researcher in evolutionary algorithms, shared some insights about how his experience at the 4th Heidelberg Laureate Forum (HLF) opened doors for him at the European Organization for Nuclear Research, or perhaps more recognizable by its acronym, CERN. As in most cases, the steps toward any destination can be convoluted and seemingly unrelated until the point of arrival.
Kamil dreamt of working at CERN since he was 12, when the Large Hadron Collider was first taking shape. Self-admittedly, he realizes that is an odd moonshot for a 12-year-old to strive for, regardless he knew he wanted to be a physicist at CERN. Though both his life and research went in different directions, the vision experienced a rebirth.
“Then it came back, in a very weird way. I can still do it, just not as a physicist. Sort of renaissance of this dream of mine in recent years,” reflected Kamil.
Part of that journey began during an Erasmus year that took him to Valencia in 2008, and Spain carved a special place in his future, largely in part to one particular influence. In that year abroad, Kamil met Professor Javier Jaén Martínez, who would have an immediate and lasting impression on his life’s trajectory. “He laid the foundation of everything I am right now,” said Kamil. “Without him there would be no HLF nor CERN, nothing.”
He returned to Poland to finish his master’s and during a brief stint working in the industry, it became painfully clear to him that this was not the direction he wanted his life to go.
“You’re in this environment during your master’s in which you solve tough, tough questions, the most elaborate algorithms and it’s very stimulating,” said Kamil, “Then you start working in industry and it really isn’t.”
“I just wanted to do something exciting, any algorithmic problem of high difficulty.”
At the time, evolutionary algorithms were experiencing a surge in popularity and a PhD position was available under none other than Professor Martínez. Kamil sprang at the chance not only to return to academia, but to Valencia as well. Towards the end of his PhD, he applied for a position at CERN, but received minimal feedback and decided it might not be where he was supposed to be after all. Then, before finishing his PhD in 2016, he read about the HLF in a newsletter from the Association for Computing Machinery (ACM) and immediately thought, “Wow! That is something that I should do.”
His expectations of what the HLF would be were not met, but surpassed. “Honestly, I was expecting to see a conference of some kind with the important people simply appearing, giving a talk and disappearing, and that would have been fine,” said Kamil, “But it was just absolutely brilliant.”
He advised future participants to be “very aware of what sort of event it is. You’re going to be interacting a lot with people you usually only read about. So be prepared in a way to have a meaningful conversation with anyone you admire.”
The HLF broke down the invisible barriers that separate the laureates from the rest of the scientific community. In Kamil’s words, “There’s this magic world between those guys and us, and all of a sudden, it’s not there anymore.” Perhaps more important to his personal development, the experience at the Forum also motivated him to try his chances again at CERN.
For his application this time around, Kamil enlisted the help of an HLF laureate, Chief Internet Evangelist at Google and AMC A.M. Turing Award recipient, Vinton Cerf. “This sort of person shouldn’t even answer an email to just a guy. Not only did he, he also wrote a recommendation letter. He’s just an outstanding human being. And all of a sudden the phone started ringing.”
After several months, the tides changed and the interview process began. This April, Kamil will become a research fellow at CERN.
His gratitude to Vint Cerf has not extinguished and as a childhood dream comes to fruition, Kamil is confident and eager to showcase his skillset. “I’m not there by accident, I feel like I’m going to do ok.”
Der Beitrag How the HLF helped to pave a path to CERN erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag PI DAY 2018: Think You Know Everything About Pi? Think Again … erschien zuerst auf Heidelberg Laureate Forum.
]]>And if they get good at that, they up the ante by entering day-long Pi recitation competitions. I can’t think of a better way to spend 10 hours, can you?
As insanely fun as that must be, I use the occasion every year to share obscure Pi trivia and Pi history facts with Pi lovers everywhere. This year, I’m also out to clear up the huge amounts of disinformation around that infinite number you get when you calculate the ration between any circle’s circumference and its diameter.
Hey, one can only take so much fake news …
But you already know everything about Pi, you say? Okay. Maybe you among the few folks around who know that the irrational number once called Archimedes’ number wasn’t invented by Archimedes at all. Archimedes merely named the number after himself and popularized it. The Babylonians, the Egyptians and even the Bible mention 3.14159.. centuries before that ancient Greek was even born sometime around 280 BC.
And perhaps you’re one of the few people around who know that it was William Jones of Wales (no known relation to Wikipedia founder Jimmy Wales) was the one to give Pi its name back in 1706. And you might also know that March 14 is both Pi Day and Albert Einstein’s birthday. Sadly, it is also now the day that Stephen Hawking passed away.
Read an in-depth history of Pi here.
Haven’t stumped you yet? Well, read on. Following is some of my favorite obscure Pi triviata and factoids. Enjoy.
And happy Pi Day 2018 from all of us here at the Heidelberg Laureate forum. Enjoy!
Can you honestly say you knew all the Pi trivia and strange facts above? If so, we salute you.
Either way, have a terrific Pi Day 2018. Don’t party too hard. It’s a week night! And please, have some Pi for me …
Reporting from Hong Kong for HLF, I’m Gina Smith.
Infographic: San Francisco Exploratorium
https://www.exploratorium.edu/pi/pi-day-history
Cover art: Timothy Edward Downs for aNewDomain
Der Beitrag PI DAY 2018: Think You Know Everything About Pi? Think Again … erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag Inspired by Star Wars, BYU’s Futuristic 3D Display Tech Works Like This … erschien zuerst auf Heidelberg Laureate Forum.
]]>But it was Iron Man hero Tony Stark who sealed the deal.
And, as deals go, this one’s huge. In a paper published in Nature recently, Smalley’s team announced it had achieved a world first: It had built a true, 3D volumetric display that, like Princess Leia’s hologram message to Obi-Wan Kenobi in Star Wars (1977), could project free-floating, 3D moving image in the middle of thin air.
Our group has a mission to take the 3D displays of science fiction and make them real,” Smalley said. “And we’ve done that.”
Smalley’s team wasn’t the first to try to replicate the Leia projection. Over the years, several research groups around the world have tried, but failed, to duplicate the holographic feat.
The problem was that everyone has been trying to use holography to mimic that holographic projection in the movie.
But one day, while watching a scene from Iron Man, Smalley noticed that holography could never be used to create the VR suit in that film.
“It was an epiphanic experience,” he said. “When Stark sticks his hand in over the lit-up table, he’s blocking the light.”
You know, until that moment, “Until that point, you know, I’d honestly thought holography could do anything.” But afterward, he knew one thing it couldn’t do. “There’s no way you can use holography to create those images,” Smalley says.
“You can’t block the light in a hologram.”
“And it suddenly occurred to me that you’d have to build this, not with one light source, but with a bunch of flying, floating nanobots, and all of them shooting lasers!”
Laser-firing nanobots? What?
Don’t laugh. The so-called optical trap display (OTD), also known as a phophoretic display, isn’t so much different from floating, laser-firing nanobots. Not in principle, anyway.
The display works by focusing an array of tiny, near-invisible lasers at a single particle — moving it rapidly through the air to create a rapid-fire illusion of an image.
Think of it as a kind of twist on an Etch-a-Sketch toy. The lasers rapidly move that light-catching particle in the desired 3D shape — and it happens so fast, the human eye takes it in like a whole image.
The result is full-color, aerial volumetric images with 10-micron image points, which appear as persistent images to anyone looking at it.
Smalley’s team’s so-called optical trap display (OTD), also known as a phophoretic display, draws images in the air in much the same way that an Etch-a-Sketch toy does, he added.
It utilizes an array of near-invisible lasers to manipulate a single tiny particle in place.
Essentially, the display is utilizing a laser beam to trap a particle. Once trapped, the laser beam moves the particle about in the air, which creates an image sort of like the one you see when kids draw shapes in the air with sparklers.
The easiest way to understand the innovation is to think of it as a 3D printer of light.
It’s not such a stretch. “You’re actually printing an object in space with these little particles,” he said. “You might think of it as a 3D printer — of light.”
“The images are rudimentary, as you can see from the videos and images in this article. But they’re likely to get a whole lot bigger and more detailed, Smalley said, pointing out that just using multiple cellulose particles (instead of one) to “draw the light” make all the difference. That parallelism in particles, he added, likely will require all sorts of light modulation and other optical tech, he says, but eventually running multiple particles in parallel should allow these displays to project larger and larger images.”
And once we get to 100 times or 1000 times what we’ve got now, I think the sky is the limit for applications.”
Smalley’s BYU-based electro-holography research group isn’t just developing various methods and techniques for creating the first ever low cost ‘holographic display,’ using the term loosely.
The team also is researching other uses for photophoretic traps like the Smalley’s OCD.
One potential use for the tech, Smalley suggests, is satellite tracking. “Human operators on Earth have to track satellites that are traveling thousands of miles per hour and along non-linear paths. They have to keep them from colliding, basically, which is stressful,” Smalley continued. “And they have to abstract all that from a regular, 2D display.
But if we could create a volumetric display with a trapped particle (matched) to each tracked object, then the satellite tracker could see — intuitively and viscerally — if two satellites are going to crash,” he added. “That would reduce collisions — as well as the cognitive load on the trackers,” he said.
Another intriguing use for the volumetric display, he added, is to build ultra large displays as projected from small devices.
“Think about mobile phone size and portability, too,” he added. “There’s always a push/pull effect for size and portability. But what if you could decouple the size of your screen with the size of your phone?” Smalley said.
“Once you decouple them, you could, say, start using your smartwatch as a peripheral to your smartphone. If you could get it the (volumetric display) to project out a sufficient large image you even could replace your phone with that watch. I’m not claiming we’ll be able to miniaturize our (OTD) design to this degree, of course. But technologies like this one, which can theoretically be used to project images far, far larger than the projecting device itself, have great promise, he said.
These are just a few examples of the sorts of pragmatic solutions that are possible with this innovation, he said. “But it isn’t quite as fun,” he admits, as chasing future tech as imagined by sci-fi.
“My quest has always been to create the Princess Leia projector .. and also I’ve long wanted to build something like the Holodeck from Star Trek, he adds. “There is great value in work that captures the imagination. Think,” he said, “about Elon Musk’s Falcon Heavy launch.”
Launching Musk’s roadster just seemed whimsical, to say the least. But when the camera started streaming the song, Starman, I found it just totally awe-inspiring.
“I suspect that single moment did more to encourage my kids to become engineers than anything I, as an encouraging parent, had managed up to that point.”
It probably goes without saying, though, that no one had to lure Smalley into the field.
“Put it this way,” he said. “I was a speaker during my high school graduation. And halfway through the talk, I ripped off my cap and gown to reveal a homemade Star Trek uniform underneath.”
“I then went on to explain that, actually, I was a Starfleet historian who’d been slung back to that moment in time, to witness that graduation,” he said.
Der Beitrag Inspired by Star Wars, BYU’s Futuristic 3D Display Tech Works Like This … erschien zuerst auf Heidelberg Laureate Forum.
]]>Der Beitrag What’s in a scientist’s mind? the HLF Questionnaire – Part 3 erschien zuerst auf Heidelberg Laureate Forum.
]]>Martin was awarded the Fields Medal in 2014, one of the biggest achievements in mathematics, for his groundbreaking work on stochastic partial differential equations. As a laureate, he tries to participate regularly in the HLF, whenever his tight schedule allows. Between chats with enthusiastic young researchers, discussions with other laureates and press conferences, he took the time to sit and talk about the big questions in life (i.e. our questionnaire), and to give his opinion on the most pressing matters for young people.
In his opinion, to do good research one should keep an open mind and not focus on solely one area. Even if learning about other subjects doesn’t lead to writing an article, one should avoid the trap of only reading a paper that is immediately useful to one’s current project. But in order to have the peace of mind to do this, one needs funding, and Martin is aware of how the current funding system doesn’t help. In his opinion, nowadays you either get a lot or you get nothing, and that’s a problem: a “rich gets richer scheme”, opposite to a “small funding scheme”, where many more could benefit from it. For Martin, it’s not completely clear how to change this, considering that the people who decide how to distribute money need to justify their budgets in front of governments, and the easiest thing is to go for big shiny projects instead of small funding scheme. The scientific community is listening to Martin Hairer, perhaps it’s time we asked the policy makers to pay attention too.
Der Beitrag What’s in a scientist’s mind? the HLF Questionnaire – Part 3 erschien zuerst auf Heidelberg Laureate Forum.
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