Advice for Research Maths

We want to extend number theory from the zero-dimensional case of numbers to the one-dimensional case of polynumbers to the two-dimensional case of maxels (or matrices). In this video we look at the family of spread polynumbers which play a central role in Rational Trigonometry, developed in my book Divine Proportions: Rational Trigonometry to Universal Geometry (Wild Egg 2005).

We'll see that the two dimensional point of view does shed new light on this family, particularly when we apply the linear algebraic concept of a matrix inverse to this on-going lower triangular maxel. As we are getting used to, the Harriot Pascal maxel is not far from sight!

An electronic version of the book Divine Proportions: Rational Trigonometry to Universal Geometry is available at wildegg.com (actually the physical book can also be obtained there).

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Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?

list=PL8403C2F0C89B1333

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list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov

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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...

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Exceptional Structures in Maths and Physics via Dynamics on Graphs" https://www.youtube.com/playlist?list...

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Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...

Algebraic Calculus One: https://www.youtube.com/playlist?list...

Algebraic Calculus Two: https://www.youtube.com/playlist?list...

0:02

hello everyone i'm Norman Wildberger thanks for joining me on this adventure we're looking at

0:07

maxel number theory extending our view of the the entire framework of numbers and how they uh

0:16

you know interact with other parts of mathematics we're trying to move towards a two-dimensional

0:22

point of view towards families of polynomials or poly numbers and we've talked about the Chebyshev

0:28

polynomials of the first kind and the second kind now it's time to move to the spread poly

0:33

numbers which i remind you are my version of the championship polynomials of the first kind and

0:40

they are i think demonstrably superior mathematically and more natural so i want to um

0:48

to look at them now from the point of view of this maxel point of view this two-dimensional

0:53

point of view and see what we find so the the standard resource for this is my book so let me

0:58

just show you here's the cover of my book divine proportions rational trigonometry to universal

1:04

geometry published in 2005 uh by my little company wild egg and if you actually want to

1:10

copy either a physical copy or an e-book copy you just go to wildege.com okay now

1:17

so let's it's probably useful just to skim down here and there's the preface okay it

1:24

introduces a remarkable new approach to trigonometry and euclidean geometry with

1:27

dramatic implications for mathematics teaching industrial applications and the direction of

1:31

mathematical research and geometry and i think that's not an exaggeration okay um and

1:41

so the first sections are about preliminary material background material and then the heart

1:46

of the matter the second section on rational trigonometry where we get to chapter eight to

1:51

the spread polynomials after having introduced the various important quantities and the the main laws

1:56

of irrational trigonometry so let's fast forward to the chapter on spread polynomials that's 101.

2:05

all right so here's the chapter eights on spread polynomials i've shown you

2:08

a few things the equal spreads formula the recursive definition of the spread polynomials

2:16

important idea that it it gives you triples of numbers that it's allowed it's allowing

2:24

you to find the spreads associated to duplicating successively equal spreads and

2:33

and then we we get the values of some of the beginning spread polynomials some graphs

2:41

and we can also work over finite fields which is quite interesting um

2:45

so that's in fact very interesting and then we have these explicit formulas some of which

2:49

i've shown you before i actually flashed this one up uh in an earlier thing but it

2:54

wasn't very clear because the r was not defined so here's the actual

2:58

complete statement involving a quadratic extension so you have an s if you have a quadratic extension

3:04

and so you get an r that satisfies this and if you have a square root of minus one or if you have a

3:09

number i such as i squared equals minus one then you have this succinct pleasant formula for the

3:14

nth spread polynomial and okay um also a note this formula here i showed you to you i think in an

3:24

earlier video i attributed it to mike Hirschhorn but i have to retract that i think uh that was

3:29

actually my my discovery um if it was mics i would have put something to that effect there

3:36

okay um some results of sushi and go the formula for uh the general

3:44

spread polynomial found both by Michael Hiershorn and sushan goh that's giving

3:49

an explicit um you know expression for it as a polynomial in dns and then here is an important

3:59

kind of variant that if you look at the scale spread polynomials okay so um this corresponds

4:06

to a scaling that we've actually seen quite a lot of times both in the in both Chebychev cases where

4:13

we sort of multiply things accordingly so what you actually do is you take uh the spread polynomial

4:18

evaluated r over four so that's some sense sort of like cutting down uh the coefficients by

4:24

factors of four and then you multiply by four to compensate a little bit and then you get um

4:30

uh the the scale spread polynomials and they can be evaluated in in a very pleasant way that

4:38

this was discovered by Mike Hurshorn okay so we're connecting yet again with the pascal matrix um in

4:48

somewhat different form but we've seen this form already where the

4:52

usual rows of the pascal matrix are sort of down these cross diagonals all right

4:57

so take this matrix here says mike and what you do is you slide the whole thing down two so that

5:04

one will will slide down over that two and you add that entry to the one that's you know two below it

5:11

so this one slides down you add it to the four to get five so um so this row here will become three

5:17

one there it is there uh this row here will become five five one there it is there um the other thing

5:25

that you have to do after you've done this sliding and adding is you you rub off the first column

5:30

all right so imagine truncating there so we're starting with this column here and when we add um

5:36

successive things then we get the success of odd numbers and then the next

5:41

uh column this one here we get the the square numbers and the observation is that if you look at

5:49

the rows of this matrix here this maxEL here that's going on then the rows that start with

5:58

the squares so every second row those are exactly the coefficients of the scaled spread polynomials

6:09

so again we see a sort of a matrix point of view the fact that we want to look at this thing

6:15

two-dimensionally and then we can perform this sort of operation sliding and so on

6:20

in in a pleasant kind of two-dimensional way and then these rows very pleasantly give us

6:25

immediately the scaled spread polynomials so for example the third one okay s3 star let's say of r

6:35

is nine r minus six r squared plus r cubed well i guess we have to um also uh uh put in some

6:46

minus signs okay so there's a nine here there's a minus six for this one and there's a an r um

6:54

cube there so this row right here corresponds to the third scaled spread polynomial

7:03

okay and there's the also the generating function due to mike here shawn and then there's a lot some

7:09

more stuff okay so uh if you are interested in the the spread polynomials i think it's great

7:15

to um to get this this chapter in fact i might uh make a separate pdf so that it's available

7:22

uh for people just want to buy the pdf instead of buying the entire book although the entire

7:27

book is only it's less than ten dollars so but in any case this is a great resource

7:33

for starting an exploration of these spread polynomials and you know as i've mentioned

7:39

there's so many important things to discover about these things in particular in the direction of

7:46

sushengo's discovery of the factorization so the the clarification of what's happening with these

7:51

spread psychoatomic or gold polynomials is a major number theoretical challenge people okay

7:58

major number theoretical challenge um that you know you can spend i'm i'm convinced you can

8:04

spend years uh investigating things and you will find more and more interesting things about them

8:11

okay so i want to reinforce what i've told you here so let's have a look at these spread

8:18

polynomials and also the scale spread polynomials you know with an explicit sort of technology of

8:22

using scientific workplace to get actually at matrices and that will allow us to go further and

8:28

exploit the linear algebra which is our basic you know aim here to try to bring the number theory

8:35

and the linear algebra together so a very quick recounting so here the spread polynomials is the

8:41

definition here is the generating function of Mike Hereshon you expand this out in the in variable t

8:47

and then the coefficients will give you the list of spread polynomials which i've

8:51

put down here okay and then we can just record all these numbers in a matrix so there's a matrix s

8:58

okay so one for the first one we're leaving out the zeroth one and then four s minus four

9:05

s squared is represented here so notice that all of them start with an s okay

9:10

so there's no constant term for any of them because all of them have the property that

9:14

when you evaluate at zero you get zero all right so we end up with the actual

9:20

the squares that has that first entry now that first column okay 149 16 etc and uh and

9:28

here then you know the rows here correspond to the various spread poly numbers okay great now

9:36

what we've done in previous cases is we've looked at such a matrix and we've observed that

9:44

well that the columns have a kind of a common factor typically the element that's sitting on

9:48

the diagonal so it was useful uh in previous situations to divide or normalize by that

9:55

diagonal entry basically sort of scaling each of the columns so that the diagonal entry

10:00

becomes a one or something or one minus one perhaps okay so let's look along the

10:05

diagonal we have powers of a four here powers of four so let's make a diagonal matrix say d

10:12

and uh with powers of four along the diagonal so then the rescaling that we're proposing

10:19

is obtained by taking this matrix s and multiplying it on the right by d inverse

10:25

that effectively divides each of these columns by the corresponding power of four and when we

10:32

do that we get this matrix r which is um the sort of scaled you know scaled the maxel uh that that

10:42

uh that corresponds in fact to this scaled spread polynomials so this scaling that we're

10:48

doing here but with multiplying by this matrix d is effectively exactly doing that kind of scaled

10:54

you know replacement that i showed you previously okay so for example here you

10:59

remember four minus one that that row and nine minus six one that corresponded to the third

11:05

scale spread poly uh number okay so that's uh that's good so we're making contact with what

11:12

we've just done before now what about uh you know so so it's this matrix here that we can obtain uh

11:20

using uh the method of my kearshorn of taking the pascal matrix sort of suitably in a checkerboard

11:26

fashion and you know shifting it down to and then disregarding that first column that was exactly

11:32

obtaining this thing here okay now i want to go a little bit further and introduce the magic of

11:41

inverses all right so what does that involve well i'm going to take this matrix r okay and look at

11:50

its inverse that's a basic thing that we're doing over and over again that we got an interesting

11:56

axle in fact this is a non-maximal it keeps on going okay so we notice that it's triangular

12:03

okay so that's that's great that means the notion of an inverse is a kind of

12:07

a well-defined notion at the level of these on maxel and then we're interested in identifying

12:13

what that inverse is and maybe it will tell us something maybe it will give us more information

12:18

because taking the inverse of even a triangular matrix is highly non-trivial operation

12:23

so it's doing something that's numerically quite involved especially the

12:27

the matrix is big and so if there's structure there that's revealing some kind of deep

12:32

um yes some deep aspect of this situation okay so what is r inverse well here it is right here

12:39

okay maybe i should show them to you both so there's r and here is our inverse

12:46

well um

12:51

what do we notice about this well so what you can do is you can

12:58

um you know in all these things with that you want to see what the pattern is okay so there's a kind

13:04

of a tried and true method which i i hope is now becoming quite sort of almost second nature to you

13:10

right now what we want to do is we want to look at this thing sort of one row or column at a time

13:15

and connect with the online encyclopedia of integer sequences which is our prime

13:20

resource for a lot of these explorations okay so we take say the first column when we we stick the

13:27

first few entries into the online encyclopedia and ask well you know does it recognize this column

13:32

and if so what is it in particular maybe it gives us a formula for it okay in this case it does

13:39

and then maybe we try the second column and then the third column and um and if you then

13:46

stare at the various formulas he's ah there's actually a pattern here i'm able to write down

13:51

a formula for the the general entry of this of this maxim and it turns out that uh

13:58

it's a surprisingly a kind of simple thing okay so have a look at this function of two things

14:05

f of i j defined by the binomial coefficient 2i choose i minus j okay then the corresponding

14:21

maxel is so in other words if i then just ask scientific workplace to conjure up

14:28

a matrix whose i j entry is given by this then it gives me this thing here

14:36

and a little observation shows that this thing is pretty well the same thing as we have here except

14:42

for this sort of familiar uh alternating kind of uh minus one uh minus one so so in this case the

14:50

the every second column is multiplied by minus one okay otherwise these two things are the same

14:59

okay so the scale spread pulley numbers okay here again they are

15:10

sort of evaluated using the mike kirshon's generating function just sort of modifying it

15:15

accordingly and then uh so here you can see what they are and and and then the claim is

15:21

that the scale spread polynomial matrix the one that we called r okay is just the inverse of the

15:28

maxel with i j entry f i j equals minus 1 to the j plus 1 2 i choose i minus j

15:38

and finally if we want to use this matrix t or the t inverse that we've just obtained

15:44

in this nice way to get a list of scale spread polynomials we can just multiply t inverse by

15:53

a column of powers of x okay and then we get this matrix here which uh it's actually just a list uh

16:01

vector whose uh whose ant entering is the spread poly number scale spread poly number and then if

16:09

we call this say s of x then we and we want the original spread poly numbers well that involves a

16:18

dilation the opposite of how we went from spread to scale so we can compute one quarter s of 4x and

16:27

that gives us then this matrix here and there we recognize the original spread poly numbers

16:34

so it's a very pleasant story that i think illustrates the power of the maxel point

16:39

of view to get at these very interesting and important i think family of pulling numbers.