Advice
A somewhat philosophical discussion about the role of pattern in pure mathematics, and then more specifically how to look for larger higher dimensional structures to understand the current objects you might be studying. We make this concrete by looking at three different manifestations of the number 28, and putting two of them together in the Pascal - Harriot matrix, or better yet maxel, or better yet on-maxel (since it is not a bounded object).
This Pascal - Harriot on-maxel is surely one of the most fundamental objects in all of mathematics.
For more on the maxel point of view towards matrix algebra, have a look at Math Foundations 165 and the videos following : https://www.youtube.com/watch?v=rL2GK...
Here are the Wild Egg Maths Playlists (some available only to Members!)
Algebraic Calculus One (for the Open Learning Course) : https://www.youtube.com/playlist?list...
Algebraic Calculus Two (for the Open Learning Course in preparation): https://www.youtube.com/playlist?list...
Advice for Mathematics Exploration (for Members only): https://www.youtube.com/playlist?list...
Solving Polynomial Equations (Members only): https://www.youtube.com/playlist?list...
Hexagrammum Mysticum (Members only): https://www.youtube.com/playlist?list...
Algebraic Calculus and de Casteljau Bezier curves (Members only): https://www.youtube.com/playlist?list...
Exceptional Structures in Maths and Physics via Dynamics on Graphs (Members only): https://www.youtube.com/playlist?list...
Maxel Inverses and Orthogonal Polynomials (for Members only): https://www.youtube.com/playlist?list...
Triangle Centres: https://www.youtube.com/watch?v=iLBGX...
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...
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Here are the Insights into Mathematics Playlists:
Maths and Music: https://www.youtube.com/playlist?list...
Diffusion Symmetry: https://www.youtube.com/playlist?list...
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Playing Go: https://www.youtube.com/playlist?list...
Year 9 Maths: https://www.youtube.com/playlist?list...
Ancient Mathematics: https://www.youtube.com/playlist?list...
Wild West Banking: https://www.youtube.com/playlist?list...
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list...
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list...
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list...
MathFoundations: https://www.youtube.com/playlist?list...
Wild Linear Algebra: https://www.youtube.com/playlist?list...
Famous Math Problems: https://www.youtube.com/playlist?list...
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...
Differential Geometry: https://www.youtube.com/playlist?list...
Algebraic Topology: https://www.youtube.com/playlist?list...
Math Seminars: https://www.youtube.com/playlist?list...
0:01
Hello everyone i'm Norman Wildberger thanks for joining me on this adventure i'm trying to
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give you some advice in the direction of doing amateur research mathematics and um i'm going
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to be talking in a sort of philosophical fashion today now you'll probably know that i'm not that
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philosophically inclined generally i like things to be concrete explicit you know on the page and
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you know abstract musings often i'm a little bit impatient with but i think it's useful to give
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you some orientation some large-scale orientation every now and then that's what i'm proposing to do
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today just to get you thinking about the structure of pure mathematics in a very general kind of way
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so these are some insights that i've gained over my decades of doing mathematical research
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and they're naturally very personal kind of insights so maybe they're not shared by other
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mathematicians but i'd like to get you thinking in terms of layers of complexity that um you know we
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can start studying something simple that's usually how we like to start and then if we examine it uh
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you know from different points of view we may find that it it fits in into some larger framework a
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larger pattern there's another layer of the onion so there's sort of these layers of the onion
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going out instead of going in which is really interesting something that we should look for and
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very specifically what i really want to do also is to move you from thinking about number theory in a
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um you know sort of arithmetical point of view to to moving to a polynomial point of view to sort of
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algebra and ultimately to a bigger picture sort of a two-dimensional point of view involving maxels
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so i'm going to touch base with maxels which are my version of matrices and i'll tell you
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a little bit about that as we go on and i'll give you some references in the description so
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there there appears to be in some sense a general a rough principle that no matter what we're
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thinking about typically we can look to embed our current investigation into some larger framework
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often in more than one way and so we get this hierarchy of increasingly interesting things
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now there's a philosophical question here about whether this increasing complexity just carries on
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i mean certainly at our level it appears to sort of appears to sort of ascend you know above and
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beyond our view but whether the actual intrinsic nature of mathematics is such that it it keeps on
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going or whether it somehow sort of closes up you know at some point
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that's an interesting question so it may in fact be that there's some bound that sort of limits
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the possible complexity of what's happening that might be connected to you know physical bounds in
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the universe after all if you want to write things down explicitly and concretely then the amount of
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space and the amount of material in the physical world is going to constrain you obviously right so
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it may be that the physical world actually also imposes limitations on the mathematical world
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um that's an interesting view we don't know okay but i think we should keep our
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our minds open to possibilities like that in fact it might even be
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although it's you know almost beyond our our comprehension but it might be that there's some
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master mathematical theory or if you like a master mathematical formula that in some sense embraces
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all of other mathematics within it now are we ever likely to be in a position to answer these
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questions probably not you know we're probably like like rabbits you know trying to understand a
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a modern aircraft carrier or something but okay we can still ask the questions and one of the aspects
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that i'm going to be talking about today is that at least from our point of view what we can think
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about is leapfrogging from objects to patterns to objects to patterns so you have an object you then
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you can put it into a pattern and that pattern itself can then be considered as a new object
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which may be just a constituent of some bigger pattern okay so patterns can create objects
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objects can be arranged in patterns and patterns can create objects and there's some kind of in
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general kind of leapfrogging that we're we're open to contemplating so that's very general okay but
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uh in fact what i what i more specifically want to do is to move you to uh to move think about
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number theory as a progression towards polynumber theory and that's a progression to maxel theory
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sort of a two-dimensional algebra and i'm going to try to illustrate that today by looking at a very
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specific object the number 28 well 28 is actually kind of an interesting number okay it's 4 times 7
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it's 1 more than 27 which is a cube that's somewhat a little bit interesting it also
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turns out to be a triangular number and that's because if you take 1 plus 2 plus 3 plus 4 plus
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5 plus 6 plus 7 okay and evaluate that you get 28. so a sum like that is called a triangular number
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and the formula for it is that it's a formula so you take 7 times 8 all over 2 okay that's the
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same thing so that's a triangular number and of course triangular numbers go back to the ancient
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Greeks here are the triangular numbers and you see 28 appearing there there's a nice formula
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for it which is basically this one here so this is one way of thinking about the number 28 not
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as an individual thing but as part of a pattern it's part of the pattern of triangular numbers
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great but you know things can be elements of lots of different patterns
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and the number 28 is also an element of another important pattern the pattern of perfect numbers
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here's the pattern for perfect numbers that's uh 396 on uh Neil Sloane's online encyclopedia
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of integer sequences which of course is a prime reference that we use all the time now perfect
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number is one which is equal to the sum of the proper divisors of k so you take all the divisors
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less than the number add them all up and if that's equal to the number then the number is perfect
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so the divisors of six which are less than six are one two and three and they add up to six
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the divisors of 28 which are less than 28 are one two four seven and fourteen and you add them
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up you get twenty eight the next one's 496. next one's eight thousand one hundred and twenty eight
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and uh then there's some some bigger ones and it was known even back to uh to Euclid that there
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was some you know intimate direct connection with these perfect numbers and so-called mercen primes
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these are special problems in the form two to the n minus 1. and of course famously all the perfect
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numbers that are known are even and we don't know whether there are any odd perfect numbers that's a
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famous problem so 28 is also in this this family but you'd have to say that this family is a lot
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more subtle and and hard to get your head around than the triangular numbers which are
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pretty straightforward there's a nice clean formula for them and the perfect numbers are
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much more complicated okay so there's another way of thinking about 28 which it comes about
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by looking at binomial coefficients if we take 1 plus x to the 8 okay and we expand that out
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we get uh this sequence here um 1 plus 8x plus 28 x squared plus 56 x cubed plus 70 x to the fourth
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so we get the binomial coefficients that you will have seen probably many times and notably
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28 appears there it also in fact appears here because it's always symmetrical so 28 is appearing
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as a binomial coefficient in fact that's not a surprise because the triangular numbers are
8:09
binomial coefficients so this this allows us now to to illustrate what i'm trying to get at is that
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once we have a pattern we can think of a pattern as being an object okay so this pattern of
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binomial coefficients this is another way of sort of embedding 28 into a pattern but the pattern
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now is not an ongoing pattern like like the triangular numbers but it's rather a very
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finite specific pattern starting with 1 8 28 56 70 56 28 8 and 1 so it goes up and down
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and once we've done that then we can sort of think of that as being an object and then we can think
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about positioning that object in a context of similar objects right so the one obvious way to
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do that is to relate this power of 1 plus x to other powers of 1 plus x okay this is just the
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n equals eight case of the more general binomial expansion of one plus x to the n
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now once we've done that what we're really doing here is is making a move from one dimensional
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kind of algebra this is like a one-dimensional thing it's a it's a polynomial or a polynomial
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okay which the way i like to think about a polynomial as you know is as a vertical
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list of perhaps sometimes a horizontal list of coefficients it's really the coefficients
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here that determines this object that list of coefficients is in some sense the object
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and so now what we can do is we can position that particular list in conjunction with
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related lists and when we do that we get sort of a list of lists or family lists which is really
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a two-dimensional object and i want to emphasize the the interest in thinking about this and trying
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to take some one-dimensional situation and expanding it into a two-dimensional situation
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this is a very very fruitful uh kind of general thing to think about
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so how can we do that well here i've i've done it okay i've put the one plus x
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to the eighth coefficients in a row right there and that's just the sort of the eighth row of the
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various rows that are appearing here these are just all the rows of the pascal triangles like
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this is the pascal triangle written as a matrix and when we think about it that way we can think
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about it as the matrix p p representing the pascal matrix because the rows are the entries of the
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powers of one plus x to the n on the other hand if you've taken the algebraic calculus
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one course or you're taking it you will know that this configuration is very important for a
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a different reason because the columns are arguably equally as important as the rows
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even though their nature is very different so the columns here i call the Harriet or basis sequences
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and we're starting with just the constant column of ones and then here just the numbers
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and then here are the triangular numbers that we've already seen and we see 28
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appearing here and then the next ones are the pyramidal numbers you get by sort of stacking
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um cannonballs in a three-dimensional pyramid uh one and then three in the base and then one on
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top that's four and then six on the bottom and three and then one that's ten etcetera and then
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higher dimensional analogs of that so of course this is just a a snapshot of this array which
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actually keeps on going and i prefer to frame all these things in the language of maxels which is
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a more data theoretic sort of approach to matrices where the actual size of the array is not really
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that crucial what's really crucial are the the multiplicities with which these entries
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appear okay so ultimately we end up thinking about such an object as really a multi-set
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okay a multi-set of pixels so that turns out to be a very powerful and pleasant wave of thinking
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all right great so now we have this two-dimensional extension of our
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in fact two different one-dimensional views of the number 28 positioned right so here is 28 in the
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column of triangular numbers that was one interpretation of 28 and here is 28 also in
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this row of the binomial coefficients involved in the expansion of one plus x to the eight
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now once we have this two-dimensional array we make contact with the algebra of maxels
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so there's something special that happens in two dimensions
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and it it's all having to do with the algebraic structure of matrices or maximals as i prefer
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and the key operation there that we have that's sort of interesting is matrix multiplication
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okay so addition is just sort of pointwise that's not particularly novel but the multiplication
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is something that's special that just happens in this two-dimensional situation in the way it does
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and that's manifested in in a lot of ways but in linear algebra what it amounts to is that
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this problem of taking inverses of matrices looms as almost essentially the central problem of of
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linear algebra and so um once we have such a thing so um i think i've defined h let me just check
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okay so this matrix h is this thing here now of course this is just going down to the
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row nine but uh in principle you know this this array can be extended okay it just sort
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of keeps on going now we're only able to view a top left corner of this ongoing array but that
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and we could do you know arithmetic with that so let's have a look at this h and i want to
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um point out a really interesting fact so an amazing fact okay
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that i will not be surprised to you if you've taken the algebraic calculus one course um
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let's calculate the inverse of this matrix okay so given a matrix a square matrix um
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probably one of the first things you want to know is is it invertible or not that's question of
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the determinant the determinant here is just the product of the diagonal and once that's going to
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be one but the inverse of the matrix is is really an interesting object almost all the time so let's
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calculate h inverse so there it is and usually the inverse of a matrix is a complicated thing
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and not obviously connected to the original matrix but we see here something really quite remarkable
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okay that the inverse of the matrix h is almost the same as h itself except that there are some
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minus ones selectively put along these sort of diagonals going down from the main diagonal
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so this first sort of sub diagonal has minus ones and then the third one does and then the fifth one
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doesn't all the odd sub diagonals get minus ones and otherwise the entries are exactly the same as
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h so of course this is um it can be phrased in terms of an identity involving binomial
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coefficients but it really sort of lives at the level of this two-dimensional matrix algebra and
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it's a remarkable fact and it's um it's telling us that it's it's a good thing to think about
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this two-dimensional point of view to think about this h as as an object like this is a useful
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and pleasant thing to do and you can tell that because the inverse is so remarkable
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okay that's sort of an extra clue that that we're on to a good thing and we're going to see that
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this approach of extending our algebra from a one-dimensional situation to a two-dimensional
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situation and then touching base with the algebra of matrices or better yet maximals is a really
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empowering kind of general principle and i'm going to be wanting to illustrate that principle
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to you by going back to some of those interesting families of poly numbers that we looked at before.