PYTHAGORAS BELIEVED THAT 'THE GODS' USED THE SMALL WHOLE NUMBERS TO DESIGN THE UNIVERSE. WHAT WAS HIS EVIDENCE?

PYTHAGORAS BELIEVED THAT ‘THE GODS’ USED THE SMALL WHOLE NUMBERS TO DESIGN THE s you read this article you may want The fortifications and organization in Samos to pause and work on some of the were extraordinary and included a military mathematics. harbour, walls around, and over, the adjoining Pythagoras hills and a tunnel right through a hill to bring Pythagoras is early amongst the Greek fresh water to the town from a distant reservoir. The harbour and the tunnel still exist. philosophers. So early that nowadays there is See, http://goo.gl/ZVMwLM debate about whether he actually existed. His dates are 2569? — 24757 BCE. This places i | Myth and fact in terms of Pythagoras early life him as one of the first of the Greek philosopher shca ee, dered lal mathematicians. The table below lists a selection of Greek Mathematicians with their dates. It is worth noticing that Pythagoras lived three hundred years before Euclid, a contemporary of Aristarchus, the first person to consider that the sun, not the earth, might be at the centre of the universe. . > . gypt Wi hi mc e pensio” next mere over to Babylon before re-appearing in Saiz about 12 years tater. Pyineiers Only Si eet Eee a 2 en oO pe a ee ai yt EE the political shenanigans. Pythagoras retired : ; The table indicates the time over which Greek to a cave, preferring to work on philosophy and geometry before moving out to Croton, mathematics developed and flourished. Some in southern Italy, where he founded the 800 years after Pythagoras there was a rich vein Pythagoreans and remained for the rest of his life. of mathematical ideas. In 250 AD Diophantus set about collecting and publishing a set of whole number problems. These problems were then worked on and shared amongst the Renaissance mathematicians leading to further work and He, So, to return to the title: ‘the Pythagoreans believed that ‘the gods’ had used the small whole numbers to design the universe’. Patterns in small numbers Imagine you are with a group of teachers at conference Pythagoras lite about 569 — about 475 BCE 1+2=3 Euclid of Alexandria about 325 BCE - about 265 BCE ... and after a pause write 4+5+6=7 +8 underneath. Aristarchus of Samos In your head construct the next line. about 310 BCE - about 230 BCE You might now have Archimedes of Syracuse 1+2=3 287 BCE - 212 BCE 4+5+6=7+8 Diophantus of Alexandria 9 +10 + 11 + 12 = 13 + 14 + 15 about 200 AD - about 284 AD ‘What do you notice?’ Have you any conjectures at this stage? Pythagoras was born in the Isle of Samos. Before reading on it is worth spending time to Samos, one of many independent rival Greek consider your own ‘noticings’. city states was ruled, at the time, by the tyrant Polycrates. Polycrates had the largest fleet in the whole Mediterranean, agreements with the Pharaoh in Egypt about mutual support in the event of an attack by Athens and so on. At Conference 2013, there was time for delegates to work both in groups and individually. When it came to ‘sharing’, ... some of the following were aired: November 2013 www.atm.org.uk eolie n

The third row adds to 42. 42 is the answer to Staying with small numbers, start with everything. » 32 + 42 = 52 Aan initial conjecture was that, because of the ... but write underneath, 42, this might be the last line that works. This conjecture was quickly abandoned! « Many participants noticed a square number 33 + 43 + 53 = 63 ... and then ask yourself what could be next? at the start of each line. In preparing for the session | had looked at this + One person noticed that the line starting with three squared has three times four before the equals sign and conjectured that the next line pair of equations asking myself if there were other triples of cubes? ... and then moving to conjecturing about fourth powers. would have four times five before the equal | had used some of this number pattern work in a sign. morning with the Birmingham branch of ATM. + The third line had three numbers after three At the end of that session Tom Francome ‘shared’ squared before the equal sign, and so on. that he had noticed a connection between the | asked participants to write down the line that odd numbers and the cubes. included the number 40 somewhere within the run, as a way of steering them to work with a Ee slightly more general special case, and as an 3+5=2 antidote to those who wanted to immediately 7+9+11=3 introduce ‘n’ into their talk. We are talking about the gods. The gods live in There was not time to work on this at conference. a higher dimension, perhaps. So, starting with | allowed it to briefly appear and thanked Tom. 1 + 2 = 3 but trying to move it up a couple of This is another pattern, | now see, like the first dimensions | wrote 1° + 2°. | notice that this adds one that is really accessible via Cuisenaire rods, Hout 3°, but 37. at least that is how | gained an insight. So, what about Prime Numbers 43 + 23 = 32 The Pythagoreans worked on Prime numbers. | want to look briefly at prime numbers from a ... followed by geometric point of view. Start by imagining a 43 + 23 + 33 = 62 collection of cubes. The Pythagoreans were ... and the next line? Is it? cubes could only be arranged in a line, whilst interested in the fact that some collections of 15 + 2° + 3° + 48= 10? © that you have made from these three ile. Ordered special cases then watch the Fu film at atm.org.uk/mt237 Watching the film suggests that the formula for the sum of the cubes is accessible to many people well before it appears in the current curriculum at early A-level. In fact both of the primary classes | worked with in Cumbria, before the conference, seemed really at home with the idea of slicing their Cuisenaire cubes up, and filling up a growing square in an organized way. others like 6 cubes, or 15 cubes could be arranged into rectangular arrays. Not many of the small numbers can be arranged in arrays, four is the first, so we need a systematic way of finding out which can, and which cannot, be so arranged. Use the grid shown in Figure 1. Start by ringing 2, and then cross out all the numbers that can be arranged in groups of two, 4, 6, 8 ... Next ring 3, and cross out all the multiples of 3. Now on to 5, ring 5 and cross out all the multiples of 5; 10, 15, 20..., even the ones that are already crossed out, watching yourself and noticing where the numbers lie. And next for 7, where do lt would be wonderful to receive feedback on any the multiples of 7 lie? Watch yourself crossing work, by learners, inspired by this type of work. out 14, 21, 28... Mathematics Teaching 237 Journal of the Association of Teachers of Mathematics

The spiral in an interactive spreadsheet, can be 3 4 “ex 5 6 SE Oe 7 UE found on-Ine at www.atm.org.uk/mt237. Circles 14 15 16 17 18 19 20 21 22 23 24 25 26 | 27 | 28 | 29 | 30 | 31 In addition to lines and areas the Pythagoreans 32_| 33 | 34 | 35 | 36 | 37 were also interested in circles. 38 39 40 “als 41 42 43 ls | #7 | 0 | a Inside a circle it is possible to draw a number of so | si | 52 | 53 | sa | 55 s6 | 57 | 58 62 | 68 | | 59 63 | 69 | 8 64 | 65 | 66 | 67 70 | 71 | 72 | 73 regular polygons as in Figure 3. 61 82 81 87 84 83 | 88 | 89 93 | 94 | 95 | 96 | 97 98 99 \ | \ / LA \ / | 100 | 101 | 102 | 103 N | i | . f | | 85 | so | 91 92 A == 74 | 75 | 76 | 77 | 78 | 79 Figure 1 Do you need to work with multiples of 11? = Using this layout the prime numbers, larger than Y 3, appear in the two columns either side of the six times table. The Pythagoreans were interested in twin primes, 17 and 19, 29 and 31 etc. Move to one of Figure 3 the spirals in Figure 2. Use the results from your first sheet to circle all the primes on one spiral sheet. A startling result is that on both sheets Of the regular polygons, squares are the most readily available so | suggest starting with a the bottom left to top right diagonal consists of square. primes. » Why? » Find a square of paper. ... Mark the centre of your square (by folding perhaps). 185 184 183 182 181 180 179 178 177 176 175 174 186/141 140 139 138 137 136 135 134 133 132 131/172 173 187 | 142|105 104 103 102 101 100 99 98 97 | 130/171 188 77 76 75 74 73 72 71 | 96 189 | 144 | 107| 78 | 57 56 55 54 53 | 70 | 95 | 128 | 169 190 | 145 | 108 3 45 44 43 | 52 | 69 | 94 | 127 | 168 191 | 146 | 109 | 80 | 59 | 46 | 41 42 | 51 | 68 | 93 | 126 | 167 192 | 147 | 110] 81 | 60 | 47 48 49 50 | 67 | 92 | 125 | 166 193|148|111| 82 | 61 62 63 64 65 66 91 | 124 | 165 reason with a cut out hexagon, or watch the film at atm.org.uk/mt237 * |143|106| Fold each vertex, in turn, to the centre forming a new quadrilateral. * Is this new quadrilateral also a square? |129|170 As it is twice the thickness of the original square is this a proof that the new square is half the & area of the first square? Now, we move on to a hexagon. 194] 149/112] 83 84 85 86 87 88 89 90 |123|164 1951150113 115 116 117 118 119 120 121 196 | 151 114 122 | 163 152 153 154 155 156 157 158 159 160 161 197 198 199 200 201 202 203 204 205 206 2071 . 162 | Figure 2 There are two routes; either cut and It is worth cutting out the hexagon, but looking at the two images in Figure 4, is it possible to agree that the triangle in the first picture is half the area of the hexagon and, by drawing in some extra ; A : triangles in the second figure that the smaller .. OF, hexagon is % of the triangle, so the smaller Does this work with other pairs of twin primes? hexagon is % of /2 of the larger one. For the hexagon the fraction is YA. These are two questions that may be worth working on at this point. Cutting out and folding gives a very different feel to this ‘proof’ November 2013 www.atm.org.uk

Figure 4a After working on these two regular The task is then to find the ratio of the three polygons consider the table in Figure 4b; sides in the triangle in the top right hand corner, Sides of regular Ratio of areas polygon http://nrich.maths.org/700 - so, if you are familiar up to the point one third of the way from the left A half hand end along the top, and search for the ratio of the three sides on that new triangle. 9 6 a problem posed by NRICH — go to Napkin fold, with that task fold the bottom right hand corner 3 4 as shown in Figure 6. | am conscious that this is A third Before reading on it would be sensible to work on both these tasks. fi Some results are shown in Figure 7. 8 9 Distance from left hand end 10 Figure 4b Triangle Y 3,4,5 4 along the top There seem to be some questions here. Y 5, 12,13 12 along the top In particular, ‘what could be a fraction for the Y 7,24,25 24 along the top pentagon?’ Ve 9,40,41 24 along the top 3,4,5 3alongthetop One reason for asking about the pentagon is And so on that the Pythagoreans used the pentagram as a symbol to recognize each other. | have But also for 24 no idea why, but one conjecture might be that they saw this as an unfinished task that real mathematicians were working on — the ratio of the smaller red pentagon to the larger one. Figure 7 The final entry raises the question, what fraction will give 5, 12, 13 the other way around?, ... that is 5 along the top. An astonishing fact, for me, is that there is a Pythagorean triple associated with every rational fraction along the top and viceversa. There is also a wonderful connecting idea between the 3,4,5; the 5,12,13; the 7, 24,25 ... ... and Pythagoras’ work on classical notes and fractions along a string. Figure 5 Pythagorean triples Because Pythagoras was in the title of this piece, More work with a square | think it is time to look at; Take a new square of paper. Mark the middle of 3? + 4? = 5? the top, and then fold one of the bottom corners What is the next Pythagorean triple that springs up to that point, see Figure 6 immediately to mind? 10 Mathematics Teaching 237 Journal of the Association of Teachers of Mathematics ltd ge m 4 PA A «i g.ù ee

At Conference 2013 there were various responses, but no-one suggested 202 + 212 = 292, which is the one that | would like to place immediately under 3,4,5. My reason for going to this triple next is that (20,21,29) is the next ‘most square’ rectangle with an integer diagonal. This raises the question; what is the next most square integer sided rectangle with an integer diagonal length? And the next? .... Development on this idea can Around a circle of radius 2 there is clearly 6,8,10, twice 3,4,5 and, from the work that has just been done, 5,12,13. But, it is also possible to put two Pythagorean triangles together, either overlapping, or side-by-side, to form a new triangle with integer sides and an in-circle whose radius we can find. As, for instance, in Figure 9 be found, in this edition of MT, as a separate article entitled Folding a Square, Page 29. In-circles Figure 9 Returning to the idea of working with circles ‘Imagine a horizontal line and cut it with a vertical line to make two corners, one above and one below.’ Now take a circle with radius one and push it gently into the top corner. Take a second The 6, 25, 29 triangle can be seen as a 20, 21,29 triangle with a 15, 20, 25 triangle removed. circle, radius two, and push it into the corner below the horizontal line. Lean a line against the two circles to make a triangle. See Figure 8. \ There is a piece of bookwork in the A-level syllabus, from many years ago, and | have just found a use for it! The bookwork states that: In any triangle the connection between radius of the in-circle and area of the triangle, with sides a, b, c is given by... Asrs where A = area of the triangle, r = radius of the in-circle and s = semi perimeter = /2 (a + b + c) The image for this is shown in Figure 10 / Figure 8 Find the lengths of the sides of the triangle. It would be helpful to work on this task before reading on. As delegates found a solution they felt at ease with, | asked them to look at a similar diagram with circles of radius 2 and 3, then 3 and 4, ..., and so on. After a little work | suggested that although there are an infinity of triangles around a unit circle | now know that there is only one integer triangle around the unit circle and this is the right angle 3,4,5 triangle. Algebra feels to be a heavy approach to proving this uniqueness. Are there any geometric proofs? Figure 10 ... from which it is reasonable to write that A = à” ua PL + Si where ris the radius of the in-circle. The formula follows. A consequence of putting this formula together with the half-base-times-height formula for the area of a triangle is that the only integer sided triangles, with an integer length in-circle radius, must also have a rational height. November 2013 www.atm.org.uk

The reasoning for this is as follows. A =rxs=axh lf, rx S=axhwe require r, s, anda... all to be integers, or at least rational, so h must also be an integer, or rational. If h is rational then it lies on a grid point and the Pythagorean triangle is assured. From this the in-circle in the 6, 25,29 triangle has radius 2. | know that there is only one integer triangle around the unit circle. This is presented as an assertion and the proof is left to the reader!. | would like to pass on a ‘wondering’ from Alan Beardon ‘/s there anything known about integer triangles round a circle of radius 27 Figure 12 The three red circles show a way of accessing Having had a number of e-mail exchanges with the connections between a 3, 4, 5 triangle and John Mason after conference we have five the associated in-circle. But, standing back, it different integer triangles round a radius 2, but is clear that if this triangle tessellates the plane, sorting out that this is all of them... is another which it does, then it should be possible to pack matter entirely. This is left open as work in the plane with circles of radius one, two and progress. three. This is a dynamic geometry task that, at conference, | left open to be worked on in Proving the uniqueness of the 3,4,5 triangle workshop time. around the unit circle is a more straightforward task. Ratio in three dimensions Multiple in-circles When you elect to set hares running at the start of a conference there are always choices to be | cannot resist showing this image next. made. | decided that there was time for one further activity. Unfortunately, | needed to set the activity up using some introductory ideas. | wanted to work on the ancient Greek geometric challenge of constructing an altar that is just twice the volume of an existing altar. In traditional ruler-and-compass work this is impossible, but by using some of the ideas from origami, and looking in the ATM publication, ‘Ideas for 6* Form Mathematics’, as a starting point, much of this is Figure 11 Here we have a 3,4,5 triangle with an incircle of radius one, and the other three circles, which touch all three sides of the triangle, having radii, two, three and six. So, a perfect result 1,2,3,6 and1+2+3=6. It leaves me conjecturing about 28, (1, 2, 4, 7,14) the next perfect number, and what the dimensions of an associated right angle triangle might be? Tessellation Figure 12 shows another way of looking at the 3,4,5 triangle. 12 opened up. | started by folding pieces of A4 paper into a cylinder in two different ways. In the first approach | joined the long lengths together to produce one cylinder, and as second approach by joining the short lengths to produce a different cylinder. Holding up the two cylinders | asked; which of these containers would hold the most Maltesers? In a straw-poll most delegates believed the ‘more squat’ cylinder would hold the most, but there were a few votes for ‘the same’, and one or two for the taller cylinder as holding the most. Mathematics Teaching 237 Journal of the Association of Teachers of Mathematics

And finally A possible different second line under 1 + 2 = 3 Instead of writing 4 + 5 + 6 underneath would it be possible to start with 2 and the next line starting with 3 Sowe get 2+3+4=4+5 3+ 4+5+6=5+6+7 Readers might like to work on the samenessses and differences between the infinity suggested by these three and the three at the start of this article. | creased both cylinders so that each was now a square based prism, and asked; will this special case preserve the ratio of the ‘volumes’ between the two paper models? References Ideas for 6th Form Mathematics, ATM Resources Linked to this article there are a number of resources on-line. www.atm.org.uk/mt237 Instructions for making a brick from a square A list of activities for delegates at conference following Geoff's plenary. Other Resources The following are web-based sources that | regret were not included in the opening conference plenary because of time restrictions. Part of Bronowski from Ascent of Man that deals with Pythagoras theorem (BBC) https://www.youtube.com/ watch?v=mOvpVOCuEdc | left the task, knowing that the side lengths were ... Or you can see the whole episode in the ratio 1:V2, for finding the volume ratio, but https://www.youtube.com/watch?v=IENM8u47zsl&play suggested that cutting a large square in half to next=1&list=PL9D6C6564285AFD14 get a pair of rectangles with lengths in the ratio Donald in Mathmagicland — golden section piece 1:2 might be a special case that could provide insight into the first, two cylinders task, maybe http://www. youtube.com/watch?v=D5n6iT2Aaqrl generalising along a chain of specialisations. ... or the whole film | next produced a brick folded from a large http://www. youtube.com/watch?feature=endscreen&N square. Then, | unfolded the brick to show the creases. | decided to finish the session with this challenge; is it possible to find the side of a square that will fold to make a brick of half the R=18v=nav0kVa66xk Archimedes volume of a sphere http://www.cut-the-knot.org/pythagoras/Archimedes. shtml volume of the original brick? http://www.youtube.com/watch?v=giUk9leseBs For instructions on how to fold, and produce, the brick go to www.atm.org.uk/mt237. There are two DVDs | often use with primary learners, although | did not do so on with the group | worked with prior to conference The dot and the line Norton Juster' http://www.youtube.com/watch?v=hgqUya0kGPA The Phantom Tollboth Geoff Faux http://www.youtube.com/watch?v=Llg5VODW6n4 November 2013 www.atm.org.uk

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