 But how could I possibly love to dance? Thank you, and thanks very much to the organisers for asking me to do this. It's a wonderful opportunity to meet up friends and a fantastic opportunity to be here and share some ideas with you. There are many amazing pictures and patterns in the natural world. This very best collogram here arrived in our vegetable processor a couple of months ago and they amazed me that each of the florets is like a copy of a whole thing, o'r bwysig, mae chyfnod ymlaen i'r adegydd yn oed, a hynny'n addysg i'n rhoi ddweudio i'w cysylltu nhw, ac mae hi yn ymbylchedd o'r ddweudio i'r adegydd yn oed, ac mae oedd yma'r cychwyn gyffredig o'r drafodaeth sgolwyd, sy'n rhaid i'ch cyhoi'r cychwyn cyffredig, a phryg o'r ddweudio i'r rhaid i'r rydydd'u cyffredig, mae ymddun o'r gweinid ychydig ynghlŷn a'r bwrdd ymlaen i'r adegydd, ychydig yn ddolch gael o'r pithau i gyfletwch yn dipyn. Tyn ni, rhywbeth eu tarfa i'r pithau i mi gallwn ychydig yn wiriwn i'r rhan ymweld ywherwydd i gyda'r byddai oherwydd i ddim yn cael ei gwaith o'r byw o'r pethau oherwydd. Fe g bailer o'r gyfletwch, oherwydd o'r fferty, oherwydd o'r mithasgrffedau, oherwydd o'r fferty, oherwydd o'r fferty, oherwydd o'r pithau i'r blwyddyn. I think it's the beauty of the patterns that promise us to keep scrutinising them, and how the pattern fits together, and I think that this work promotes that inquiry and shape dialogue. Before long, we're almost playing the hit work about the concepts and relationships as we study these patterns and how they fit together. I'm looking at many patterns over time. I think it's almost as if they're going to find a universal language that transcends time and culture down the generations. So what I'd like to do is to share some images with you, some historical and some of the thing, which I feel illustrates the art of representation. I went to the Cementions examples, and then, how Leonardo uses his artistic talent to promote that kind of view. I'd then like to do some kind of pattern, which would inspire me to think about learning in the classroom, and of the time for working on some related tasks. I'd like to be first of all to share with you my earliest reflections of representation. I must remember the time after I was planning and having had some ideas in the very basic sense, and how this was only brought on by looking at the family clock, which is working present from my own grandparents to my parents. So, this is the clock, and when I was coming up to six of my sisters coming up to nine, I remember looking at this clock and thinking about our ages on it, and I then remember thinking, well, as the clock hands go round, what age will I be there? And then, so going down further, which is really cool, I think he is thinking, well, what age will I be when he's on the one? What about when he's back on the one? I think, right, mystically, I have myself going round the clock eight times, and I'm not quite sure why we get to be that age, but I certainly wasn't at the time. And one thing I think for me was that this rose up and up. I'd added this cave to me into the mathematics around me, as I really amused myself in the most boring of the situations that the city of Bindham is opening. And I think that what's interesting here is that not only can mathematics give rise to representation, when we want to think mathematically, we want to come up with representations for each other, but also how representation in the rise to mathematics, and this works certainly on children, even though they don't really articulate it too much with adults at the time. I'm going to begin by looking at two well-known examples of great numbers from the old Babylonian period. I think this period is very fascinating, about 1600s to 1800 BC, at the end of the time, recently, in the Great East Asia, often called the cradle of civilisation for everything that happened there, and you see ancient civilisation, of course, described in the Bible, but I've been corresponding to it a long day back. So this hub, sorry it hasn't come out that well here, but it's in the British Museum, and it's commissioned with the interview on 15285, and it shows that it's small problems, and they can't see that well there from the back, they're all shown and inscribed in squares, and the idea there is to find the areas with a whole set of small problems, and the next topic that I'm going to show, there is some very possibility that the next topic will actually be the solution to one of these. So the next topic I'm going to show is the famous YBC7209 topic from the young Babylonian collection, and this one is called to have been produced by a trained scribe because the influence on it are quite large at the time that's how they've been trained. It's in a uniform script, and we can see that the wedge shapes are either going down with a less than sign, or they're more upright. It's a set of decimal notation, which is sort of grade 60, it's got a subface of 10, so each of the group of marbles represents a pg up to 59, and the little less than signs, the wide wedge shapes are like 10s, and the upright and narrow ones are like 1s. So we can have a group of up to 5 with the less than sign of the infinity, and 9 of the level up ones, making a group in there of 59. So looking at the side of the square, that shows 30, I think most of it's going to say 30 there, although that has been some sort of, possibly if it's not 0.5, because there's a decimal point there as such, so it could be 30 to 60 if it's not 0.5. Going across, apparently, no one would have had a kind of a remit gap in the line of the politicians, which they would have referred to for that ratio of the cycling to the diagonal of the square. You can see here, looking at the wedge shapes that are currently 1, the 30 for 60 if you like, or would have been perfectly first person to place, then 51 over 60 square, you can take over 60 cubed and so on. Give a really good approximation. And then underneath here, we've got to find that inside length. We've got that approximation there, what the length can be on that particular square. It's quite interesting that if we take out as not 0.5, then the root 2 and then the 1 over would actually be reciprocal. Of course, it could easily be 30 which is more likely interpretation, 30 under 30 times root 2. On this one, I'm not so fascinated by how they are going to try to understand how so many things are linked together and how he uses his understanding of art to try to put his understanding into plenties. I've been very fortunate to have spoken at the time that Martin King showed here and put it at his clock, where I showed him of my way of life and my family brought there. He's become a real world-renowned scholar on their logo, he's written many, many books, and these are two of the ones I've been looking at. The top one I've tried, it's fascinating to be a friend with you where he looks hugely at the different aspects of what their logo is looking at, so there's science and representation and so on. At this bottom line, I'm very fortunate to have that with me because it's interesting to see that. He said that the way in for him was a little bit more like the punk power of the picture I showed you. He used natural science as a Cambridge and that was the way in for him was looking at scientific images and naturally living into art. So what really interests me as a slave of their logo is how he accomplished so much in terms of painting and sculpture, architecture, another anatomy, geometry and science. But it's what he does in terms of representation, which means he's working on mathematics that I find particularly interesting. So he was really interesting in how geometry and anatomy come together and he looked at the work with the truthies of the first century Roman architects who wrote about taking the books and he looked at the third of these books where he looked at the proportions of the human body and how it just ought to be reflected in the buildings and the temples. The truthiest rights about a man and how he can have the hands and the feet positioned in a circle. But it's only Leonardo that puts both images in the same place. And it's only Leonardo that effectively dissipates the senses. Other people of time, even though the truth was didn't actually surface, had assumed that you had to have the same sense of the both. But Leonardo dissipates those and in dissipating those he could have the body in perfect proportion and actually sit in beautifully within that circle. It interests me when we actually think about the circles we could trace out if he were to move with his hands in his legs so if he moves with his arm that the fork from here is at the shoulder and that's actually going to move away from the circle so that he moves that arm. And the same in his legs. Because of course the radius of those circles is different from the radius of the circle that we see here. In terms of the square of the circle by construction to come up with a circle and a square in the same area was a problem that was going to lose it. Leonardo was extremely interested in this and shows lots of diagrams where effectively he tries to kind of produce a curve in his shoulder and actually tries to get that map on to a curve rectally in his shoulder. It was a thinking thought that he was going to break the problem down now that he might get there in the end. The last part of the canon he thought he cracked the problem but obviously it didn't remain leased in there. It was interesting in the right of pyros for that 1650 BC when it was copied from the original couple of years earlier but they thought that if they cut about a ninth of the diameter of the circle they could get to the side length of the square in the same area and that's a reason for the approximation. Lindman proved that the problem by construction is impossible but a lovely mathematician, the manager he got an incredibly close approximation which I think would make about a quarter of each out on a circle of about 144,000 square miles in only a quarter of each out on the side length of the square. Leonardo spent some years living with a mathematician called Pachiovi. Pachiovi was a tutor of mathematics and Leonardo up until that point had been really quite an interesting but actually became really extremely proficient mathematically after the best time I've been working together especially in geometry. Martin Peck was saying that Leonardo actually found any other break was really quite distasteful because he liked the kind of mathematics he could really get his hands on. He produced some drawings here of Pachiovi's book and I think it's really interesting how he used forward the kind of representation that was possible at the time in what he could actually see through the faces here of the polygner. So he did a number of these drawings called fenestrated after the French word for window Leonardo himself called vacuous. I think it's written in Pachiovi's book there were a number of letters that were done these were all in woodcut for the I don't know if you can show it in woodcut and the N is in the next and he used to say an emblem until just recently and again once again bringing Pachiovi's work with the greatest art in Western chemistry and the largest museum in Western history of art bringing the two together once again. So that representation was the starting point for Leonardo and he didn't just stop when Pachiovi left him when he could introduce his beautiful drawing for him but he goes on himself to think about working on a volume of the go-deck medium he takes one of the faces at the base of the pyramid and it's thinking you can split it up and even when he's got one of those again he takes that face and actually splits that through again trying to break it down into parts that you can get a handle on I think there might be a little bit of jubilee over the last couple of stages there but he's certainly trying to break it down It fascinates me again that there's so interesting representation which is such a key aspect of what any mathematician would be interested in and how many godswats would have different volumes Even in a basic sense that I'm going to get by over there I can see them at the thoughts at the bottom how you can take one face and actually take a beautiful volume by thinking about the pyramid I've taken the points out from there and the relationship between the the length of the base and the length of the pyramid and again you can take it again from a different base under the year everyone thoughts are going on there He also looked at Pythagoras and what Pythagoras was called for cubes I think it's interesting to think what two pieces or three pieces shapes Pythagoras would work for In fact the first time I met John Mason I was in Cambridge on the hill there and invited him in and the first thing that he asked everyone to look at again his relationships with Pythagoras So what's interesting here is that he can actually see from his juttings that he's tried four cubes there and he's tried three cubes and you can see that they add up to 91 not 135 but it would be quite a five cube and he tries going through all of the different shapes and so on and of course what we need to look at in shape when the area of the area or the volume of the volume of the volumes is proportional to the square of the cyclop so anything proportional to the square of the cyclop is going to be the tree and anything else of course won't So looking at this painting this was only recently authenticated as Leonardo some at the top of London and Pythagoras saved the world and this was sold in 1958 but just 45 count and authenticated as Leonardo to a Russian investor for 107.5 million Pantythe was just talking about having these kinds of parts of play here and he says it had that kind of presence that Leonardo was having that kind of strangeness that the later Leonardo pays his manifest he says that Leonardo's understanding of anatomy was so amazing and how he could really see underneath his sort of spirit how he could be knuckled in the world and so on that it had to be Leonardo because it said that he had that incredible understanding and also there that we need to look at that the way that the hair is done and so on It was interesting with Leonardo he backed to Santa Maria Florence and they had a hospital there where he was able to do some dissections and in doing those dissections he got an incredibly good understanding of the human body and anatomy which was incredibly unusual and beyond his time there these Islamic patterns were coming into Venice from the port from Turkey and Leonardo was very interested in these patterns and again he was particularly interested in his painting looking at the stole you can see some of this pattern work that he's produced here and why he's extended those patterns so he doesn't show up quite well at the back so in terms of what makes Leonardo such a massive representation I think for me is the fact that he tries to put what he understands into a painting it's almost as if he's actually trying to reconstruct reality rather than actually trying to look at what reality shows and then try to represent it he's studying the physics of Dray through sections of looking at how material unfolds and how we can look at different colours and what those colours would look like in different lines and so he would be able to predict what it should be doing under those light conditions and the way the material is unfolded and so on and as I say his understanding of the anatomy was so acceptance enough and the dissections that he did but really what's interesting here is just how much he tried to actually put in terms of his understanding into a painting and Marty Pepp would say that he actually tries to go beyond what should go into your form in terms of the picture he's able to hold and that nowadays people like you are selective in what they might be trying to show but I think part of it is that how many people would be able to go beyond that way to have such an understanding or so many disciplines that they could try and show for the better times so what I'd like to know is to show you a few patterns that I've found on my drawings and then looking at some of the ones that involve actually breaking up shapes into other shapes because those might be the mathematical fodder if you know the use of the classroom so the golden toys are a really rich source of mathematical information and I think it's interesting here looking at the right hand one of paint just how much overlap the other pattern has with the one on the left and inside the squares those repeating patterns to me they're very missing of all given the elongra they're really quite similar there and so looking at these examples I'm chosen to be very simple a pattern to be a hug but there are so many far more complicated ones and so this was built through Islamic rule in the 13th and 15th century it was a fortress in Valadr in space and you've got a dynamic art on all surfaces because of course they would be showing you human form well the idea is the symmetric of these two patterns and how if I look at just the blue how the blue maps on just within the blue and the white within just the right but how would so I might have got a transformation and white to the blue or blue to the white and the end here we've got a pattern there alone these patterns are of particular interest to me all made with them from the drivers here because they're actually beginning to show a great enough of an area into other shapes and this is breaking up as the same which I've been providing a fantastic opportunity for working in the classroom I'm looking at this occulting pattern I saw in Stonia if I look at the white square and I can see the tilted square I like to be in the white square if I imagine that tilted square sitting on top of the white square I can imagine folding the corners over into the centre that will show me in a practical sense that the white square is double the area of the square this is inside of course I could be a little bit more formal than that and I could use my factors on the side lengths of the tilted square to find this diagonal to confirm that root to ratio there inside lengths and one to two ratio in the areas if I don't have a square that is a much darker square this is a pwty fan of the dark square outside and I can see there in the dark square that the side length of the dark square is double the side length of the white square and I can then look at the relationships there throughout that pattern in terms of lengths and areas I like to offer a pwty because sometimes when I look at this pattern my eyes look at the top two thirds of the pattern here and when I look at the top two thirds of it I can see that I've got two big squares side by side and then in that I've got two tilted squares and when I draw back to the YBC 72.89 I think it's that root to ratio there and if I look back at the bottom two thirds it's like a wheel of the way I'm looking at it I can see I've got black around and four triangles and it's the same thing that I got to go back that way but it's interesting that each of those big squares is still made up of it this is what changed me to be this image that I first saw San Clemente Roam to me it was just mathematics buzzing up on the floor I couldn't understand where the people were looking at the walls and looking at anything else around there once I saw that that was part of an hour my poor family is I've got the camera and I'm just photographing around I just think that's beautiful it's timeless and it's beautiful I found out later that this was made for a row of ruins by the Cosmarty Artists in about the 12th century and how the darker colours are much rarer so when you look at a darker strip there are times chopping up being on much of our food and having had to break the pattern down to use less of those darker rarer colours and it is this redding up of the Cosmarty world which I just find so fascinating and so lovely and to say when I look back to the patterns that could be a hunger I love them but for me this is the work for me for the classroom because it's this breaking up which has made this us to work on the footprints so I'd like to have a quick look at what I've done from what I've done using the classroom so you've bought some of these on your table to which you'll have a look at them and these patterns are inspired by the Cosmarty worlds I'll just show you I have thrown a couple of other shapes in there just to look at the places and other combinations that we can do with the equivalents so the way that we are is that we can actually look at our shape and make up our other shapes and we can have some sort of checking if we can cover an area because we've learned that the areas must be equivalent this enables us to look at connectivity so they make the students to be creating in that they can actually make patterns of their own and they can follow the chance that I would be acquiring they've got a shape dialogue they've got a shape set to look at this is one of the tasks in getting a set of new national curriculum I want to show you another shape it's about the vertices of the new shape coinciding with the vertices of the old shape don't say that they're inscribed within the shape it's like the inverse of the shape of the new shape on your tables you've got some little plastic bags there's also a figure of plastic the figure of plastic you've got you've got plastic bags to take back to school the little plastic ones in the polyde bag you can do it any way you like if you really want to keep the plastic of the school I'll go back and I'll make my sources about plastic bags to school first you've got to be both at the very end of the course so the idea is to just have a play be creative and just put a little bit of your fashion at the red squares and the green square and when I'm talking about the green square I'm imagining of course the whole of the green square underneath the red squares however I sign with the X I'm looking at the red square obviously that's got area X and the red square is four times that area so it's four X but also how if I look at the green square and say that I'm signing for that it's two X then of course it's a representation between me and the two X four square there's been quite some discussion here about the areas of the triangles and the equivalent between areas I'm looking at the red square there how four of those red squares make up the green square and how I can see with the the red triangle up in there that four of those will make up the square but also four of the other triangle will also make it and confirm this at a range of institutions and students to actually confirm that they're at a range of interventions I look at this red triangle here the two X times X divided by two gives me X square and the same of course with the the orange one the two X times X divided by two also gives me a square and if I look at the arrow head I can see that the arrow head is a quarter of the shape is each of the others a quarter but even if I don't believe that I've got the triangle there with a base here of X and height of X X times X is X square divided by two half X square and I've got two of them building up to my X square here and so it doesn't matter how I work within that shape I want to be able to confirm that the sums of the areas are working and the equivalent areas are working okay there I like this to be shown sort of number 10 that's not number 10 down in the street of course even if that's not that far away they'll never be able to get that close I've just got about the power of visualisation and looking at something as simple as these windows I'm looking at the window again on the left most children would look at that and although it's not that they might say well I've got about three quarters of it shaded and then of course if I look at the bird to the bar coming down you might say well I've got now and so you can now see that I've got the six out event, I've got six eights and I can see the equivalence there because the amount of shading and the total amount doesn't change when I consider the birds to the bar but this one just when these windows I'm just going to give you a couple of minutes wherever you might want to just look at those images again in London and just think about how you might use those images to explore fractions also looking at the women expressions it might just give you a couple of minutes perhaps to work on that and it might have also got expressions again when you look at that with the costumers type of colours if I look at them on the right I can say I've got side eights there I've got three x by three x is nine x squared and of course how I might rearrange a nine x squared for different x by nine x and so on so more of patterns from everyday life which of course we can use in a similar way to the mathematics expressions one of them is just a simple marquee which again I thought was quite useful I love these ones in the prior and I call these the envelope patterns and if I'm looking at that envelope and I try to say what is it that makes it that sort of pattern so I've got to have the horizontal line and then the lines go on up to the corners so if I'm going to create that how much of that can I generalize so I might say I've got a specific length, a length length here or I might give it a letter but it's also a lot in the middle what might go up to the angles and so on and I like looking at the some of these unpaired though and even all of these lines and triangles that are formed within that show is finding quite difficult so really we know what's going on when I'm iniclinating the quadratic expressions and factorizing in terms of vector and facturising and having some sort of images what's going on gyda ni bod gallwn ni wedi bwysig ar gyfer y telflaeth i chi yn ei wneud fel eich cynnig. F fått y mynd i wneud llawer yn d 나올 yn hyn sy'n cael cael mae hi ar wahanol. Mae'r Dweud os ym Mhaz Blwyd, fel Gwmpalae dda, byddai'r Dweud ffyrdd, a rhaed i'r Llyfr AM himselfol. Mae'r ideaeth yllaf yn eu bod unrhyw chi rhaed i ddau'r gwahogau a'r gwahogau hyd yn ddefnyddio, A hynny'n defnyddio fath yn yra meddwl. Ieith yw'i gwneud ffodol fydd i'n cyfnod. Felly i'n gwneud amי�i fan lebu bod yn siarad, mae gynny'n gweithio mewn ddefnyddio mewn ddefnyddio'n gweithu. Felly, yn fdweithio ddau'r hyn, fe ddechrau 1 yn x plus 1 ym Ym homo. Rwy'n ei ddim yn ychydig ychydig ar ei ddefnyddio mewn ddefnyddio mewn ddefnyddio. Felly, hefyd fynd i gydig unig iddyn nhw'n gweithio'n gyda'i'r myneddau. I've got green x square, I've got two of the circle one by x's, I've got one so I've got x square down to x and one. I can present this to somebody else in an unfacturised form or in higher pieces and they may then need to go back to build a rectangle. And that's something students of course can solve for one another. It's another example of representations where we may move away from those exact 10 faces or more freehand style. And looking for some of the structures that we've just been discussing, x by x, x by 2x, looking on the right-hand side, x times by x plus 2 here, and I certainly found that students often confused 2x with x at 2 and they tend not to think as much as they want to about that bottom length if that was to say 2.5, if that was 5 out of 2 and so on. And they can work with things like this in a particular, with a particular value for x or in the generalisation shown here. If they are looking at the overall rectangle on the right-hand side, you can see we've got x times x at 2. If I look at the two separate rectangles, I've got an x square down to x again reinforcing the equivalence between the facturised and the unfacturised form. The one at the bottom is one that we've been discussing today and two different ways of showing 2x on the square. Looking at third-party representations, of course we can go further than that and collect up 3x at 4 and 2x, looking at the constant dimension x square in order to get to stack up those rectangles with the constant dimension and I can see then the 1 by 4 on the right-hand side. Where I think the cognitive conflict comes in is that if I move that green 4 by 1, I move it from here across to the left-hand side. It's difficult for someone to say that they think the areas are different. I've meant if I ask the students to work out 4 plus 5x, they may end up working out 9x, not 4 plus 5x, by supplying their operations. I feel that this idea of moving the 4 there and asking for an expression is more likely to give students what you think about that those two must be the same and to actually concentrate on the correct form of operations that is going on there. This is a basic area we want for solving an equation. Again, you can see here very self-explanatory that 5x out of 3 is 38 and I think you need to take off that little rectangle on the top and go back to the main rectangle. You can reason that the main rectangle has got a smaller area when you start it with and then go down to 5 in the side there of that little rectangle with the solution for the equation. So the principle of the area is shaking out to go back to my rectangle, adding in that extra 3, filling out the room in usually a bigger area than the area you started with and again finding out the inside of the rectangle from that. This, in terms of an equation with no number on both sides, this is sometimes called the quadratic cut between everything about your row because the student cannot avoid operating down on the air, they cannot simply do this unless it's a really true example of arithmetic substitution. They're going to have to engage in the unknown and certainly integrate themselves as complicated enough. And so the idea here is to subtract the largest possible rectangle from both and so if I get rid of the 3x rectangle from both, that's then got me back into the kind of equation that students buy much more from me into the 2x, 3x, 4x and so on. If you want to deal with something like 31 minus 3x, of course you're shaking out about the 3x to begin with and you've got to fill that hole back in adding in to the right-hand side and adding in to the left-hand side. And again, getting you back first, tricky step back to an equation that students have got to more comfortable with. Of course the idea is to be able to generate your own examples of what's wrong with that number and deserve the creativity to come in and students can then have some ownership of the difficulty that they're working with. The next thing on our back is how to play with it. It's actually looking at some base 3 property in the name or in the other message that you would like to encode. So the idea here is that 3 beads, and you've got parts of beads on your table, 3 beads will make up a letter of the alphabet on base 3 according to the position of the letter. And then if you want to change those codes for M, I know there's something around this room that's way too basic and they want to do something different. Feel free. So just showing you like this with A is 1, B is 2, C is 3 and so on. I'm looking at it in base 3 arithmetic. I've shown you the first ones there. The first basic there is my name. It's Jackie. So J is in position 10. So I've chosen here. The grade for me is 1. So I've got 1 lot of the 9. I've got number of the 3s. And I've got 1 lot of the 1. So looking at the place value, I've got 3 to the north, 3 to the bottom of the 3 square. So 1, 3, 9, 1 of the 9, 1 of the 1 makes 10. J, then the A, and then the C is the third one there. And then K. So J, K, and so on. So here, if I'm looking at 1 of the 9 and 1 of the 1 is 10, I've got 1, 1, 1, 1, 2 of the 1 and so on. So I've got my grade is 1, my ground is 0, and my black is 2. So the idea is that if you choose for you what colour you want for each bead, just make sure you stick to the same colour. And that's my middle name there. So it's going to work out what my middle name is. Except for ours, you know me. My middle name. It's only going to crack my middle name. I've got some string. It runs quite well, once you've chosen the colour for each bead. So choosing a bead cut for 0, for 1 and 2. You might have a whole round of bead that you want to have a look of, and you need my colour to go. You should have about half a metre of the colour of the grey of each. It works well if after every letter you do a knot. The black beads might sail over the knot and you may go and gut them off. There's paper clips in case the holes in them aren't quite big but there's paper clips there and half grey. This is the spell on screen. It really slows over great. You can take off, representation means grass and I think it's a coach of how we have relationships. So you can actually see them. You can't help but see mathematics built into that. Maths and students build their own structures by understanding, and that's very much the learning you can do in the classroom how you can develop their own structures by understanding as they scrutinise these colours that they discuss, how they fit together and creating a pattern that that does facilitate the study of a network of mathematical concepts of relationships. I'm just going to gloss over the main situations, provide a wealth of material that we can look at. We don't need to look that far to find it as soon as we can as we were looking at that earlier. We'd like to acknowledge anyone who has helped me and are far more people who have helped me along my journey of playing mathematics over the last however many years. So thank you very much to all the people who have participated in this and great people who have been healthy on my journey to playing mathematics.