Advice for prospective research mathematicians
Here is a quick introduction / review of the essentials of Rational Trigonometry suitable for high school or first year uni students.
Our ultimate aim here is to explain the important spread polynomials / polynumbers, which are more pleasant variants of the Chebyshev polynomials of the first kind.
We cover the five main laws of rational trig: Pythagoras theorem, the Triple quad formula, the Cross law, the Spread law and the Triple spread formula. And we give self contained proofs of all of them.
The treatment is concise, relying on a prior theory of linear algebra and dot / scalar / inner products on a linear or vector space. This is really the proper starting point for an affine theory of metrical geometry, and it naturally incorporates the geometry of special relativity. So this theory should be of particular interest to physicists.
This is part of a series on Advice for prospective research mathematicians, mostly in the amateur direction, but this topic is quite important for professionals as well, as the implications of understanding Rational Trig are far and wide in contemporary geometry as well as mathematics education.
My book Divine Proportions: Rational Trigonometry to Universal Geometry (2005) which lays out the foundations of this exciting new development is available at https://www.wildegg.com/ in either e-format or as a physical hrad copy. At this website you can also find screenshot pdf's for many of my video series!
Find a lot more videos about this two dimensional number theoretic approach to important families of orthogonal polynomials in the Members section of the channel at the playlist:
Advice for Mathematics Exploration (for Members only): https://www.youtube.com/playlist?list...
Here are the other 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...
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...
0:02
Hello everyone i'm Norman Wildberger and thanks for joining me on this adventure i'm giving you some advice about research mathematics and we've been talking about the role of maximal number
0:13
theory i've been trying to get you to think more broadly about what number theory means moving towards understanding some of the special orthogonal polynomials like Chebyshev polynomials
0:23
uh in a sort of a matrix or maxel point of view so the next step in that is to look at
0:29
the spread polynomials which arguably and in my opinion certainly are more
0:35
fundamental than the chebyshev polynomials of the first kind which they essentially replace
0:40
but in order to really make sense of that i really need to tell you a little bit about the spread polynomials and where they come from that's actually quite an involved story but very
0:49
interesting and very important so i'm going to take this opportunity to give you a very brief but
0:54
actually reasonably comprehensive introduction to rational trigonometry with a view of
1:02
introducing the spread polynomials so rational trigonometry okay has a lot of advantages
1:10
and it's i think really important to uh for you as amateur research mathematicians to
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to move towards understanding rational trigonology and the key role that that plays in terms of
1:22
understanding the framework of modern geometry so when we're learning trigonometry in school
1:30
there are essentially two triangles that are basically used all the time in various variants
1:38
that's these triangles here okay this is sometimes called the 1 1 root 2 triangle or maybe the 45 45
1:45
90 triangle and here's an equilateral triangle and if you uh bisect the base here with a line
1:52
an altitude then you get what's often called the one two root three triangle or the 90 60 30
1:58
triangle okay in terms of angles so it turns out that modern trigonometry despite all the hoopla
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is actually quite quite weak computationally and as a result of that whenever students in
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high school or even undergraduate courses are asked to actually make computations in
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the subject inevitably when i say inevitably i'm talking about 99 of the time at least
2:25
the examples that are given are chosen using these two triangles and putting them in various places
2:32
you know they rotate them around in various sectors to make it look like you know there's a variety of examples but ultimately students are always end up being asked to calculate cosines or
2:43
sines or tans of 30 degrees 45 degrees or 60 degrees or some some closely allied rotational
2:50
analog so it's actually really dramatically surprising how limited this trigonometry really is
2:59
and we get a sense of the power of rational trigonometry by looking at what happens to
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these triangles in rational trigonometry so there are no more any irrational quantities
3:10
floating around neither in terms of lengths like square root of two those disappear and also in
3:17
terms of angles so instead of having angles of pi by 4 or pi by 3 involving this infinite decimal pi
3:26
that we actually are not capable of writing down okay so all the quantities that you see here
3:33
are completely rational here there's a quadrances of one one and two spreads of a half half and one and a quadrant of four quadri is some kind of analog of the areas
3:42
minus the squared area times a factor and over here quadrances of one spreads of three quarters
3:51
and quadra of three so the fundamental quantities in rational trigonometry are not
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distance and angle and area so much as quadrants spread and quadra
4:04
all right so rational trigonometry was developed in 2005 and uh it in my book
4:12
called divine proportions rational trigonometry to universal geometry and it gives a much simpler
4:18
and more powerful and accurate theory and the computations uh run demonstrably faster
4:24
and can often be full precision so it's it's pretty obviously a superior theory but of course
4:30
it's going against just these decades of orthodoxy and and vested interest in the status quo okay so
4:38
you know it's a big hurdle to climb to try to say look guys there's a really fundamental theory here
4:45
we need to replace it with something that's a lot better even if that that theory is a lot better it's going to meet a lot of resistance
4:54
so what are the benefits well first of all you move towards universal geometry so metrical geometry becomes much more general you can deal with a general bilinear forms
5:04
and in particular for a physicist you get new understandings of relativistic geometry because
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ultimately the same trigonometry that works in Euclidean geometry works in relativistic Einstein Mankowski geometry too once you set things up correctly
5:21
so that's a very powerful tool for physicists you get chromogeometry which you will know
5:28
perhaps is a remarkable three-fold symmetry and planar geometry that opens up all kinds of
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new doors for an investigation and we get an attractive and much more general approach to
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hyperbolic geometry as well as spherical and elliptic geometry so those entire subjects
5:44
have now been transformed from this rational trigonometry point of view and if you watch my various series you can find out all about all about hyperbolic geometry for example
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and one of the other things that it does is it sort of theoretically very clearly separates the geometry of triangles from this separate issue of uniform motion around a circle the
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modern undergraduate student somehow thinks that these two subjects are all part of the
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same thing because that's how you know that's how the subject has been framed oh you've got
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you know two lines here how are we gonna measure the separation of those lines oh we're gonna we're gonna draw a circle and then we're going to think about uniform motion around that circle to
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measure how far those lines are separated okay so it turns out that we do not have to do that
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in fact you don't really want to do that and then there's vector trigonometry which is uh a variant
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of rational trigonometry that's especially useful for for dynamics and physics um i'll talk about
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that in some other point and i think really the killer application is that you know we now have
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the possibility of dramatically simplifying high school mass education i mean trigonometry is a
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stumbling block for so many generations of of students you know all these formulas cosine sine
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secant cosecant and they're inverse functions and um you know all these various relations they're
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all these sort of identities that they're supposed to uh to learn and the the honest reality is that
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in fact to develop this thing sort of logically coherently you really need a prior theory of calculus okay so this crucial fact is is carefully kept away from students and so they're made to
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pretend as if this is a kind of elementary theory but actually logically you require a theory of calculus in order to find these arc lengths of circles upon which the angular theory is based
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i'm sorry but that's the way it is so it's kind of a logical cheat and i think as a result of that
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it's not a surprise that many students are sort of alienated from mathematics because of it okay so
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rational trigonometry i'm going to give you a very concise uh approach to it using a
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prior theory of linear algebra okay so i'm going to assume that you know something about
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vectors and in particular dot products or inner products or scalar products on a
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vector or linear space so we'll stick with two dimensions uh but it works in
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three dimensions and higher dimensions too so the starting point is a dot product let's say
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for example you have a vector x1 y1 another vector x2 y2 the dot product is the number x1 x2 plus y1
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y2 that's the Euclidean familiar dot product but if you replace the plus sign with a minus sign
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product that's more in tune with with physics and relativistic physics and in three dimensions well
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there's all kinds of variants but for example uh the dot product of these vectors might be x1 x2 plus y1 y2 plus z1z2 that would be a Euclidean inversion if we replace the plus sign here with
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a minus sign we'd get some relativistic version okay so everything rests on this
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initial dot product okay then the first key definition is the quadrance of a vector a measure
9:08
of how big the vector is and the definition is that the quadrance of a vector v is just v dot v
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now it's totally remarkable that it took till 2005 for someone to say hey guys maybe this
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is an important quantity and maybe it should have a name of its own up till that time people
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call it something like oh that's the squared length which is just like you know mind-bogglingly silly because the length of the vector is obtained by taking the square
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root of the quadrants so the quadrant really comes first that's what you can actually compute and then if you can take a square root which most of the time you can't then you end up with
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a length but you see the length is secondary to the quantity the quantity is the principal
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uh theoretical and computational quantity so clearly it should have a name and then
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the quadrants between points a1 and a2 say in the plane or in space it's just the quadrants of the
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vector between them a1 a2 and it doesn't matter if it's in this direction or this direction because
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uh if you replace v with minus v the quadrants doesn't change so for example in the Euclidean
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plane we get the usual formula for the quadrants between two points okay the quantity between the
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point x1 y1 and the point x2 y2 is x2 minus x1 squared plus y2 minus y1 squared and we are not
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going to start talking about the square root of that quantity and so the distance we're going to
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follow euclid okay who carefully avoided talking about in fact distance and angles in the elements
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contrary to the usual idea that Euclidean geometry is based on distance and angle i mean that's very far from the truth
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okay and then there are the two main theorems in geometry which are in fact also in my view the two
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most important theorems in all of mathematics the first is pythagoras theorem which concerns
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three points in the plane a1 a2 a3 and the perpendicularity between these two sides between
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a1 a3 and a2 a3 and the statement is that these two sides or these two lines are perpendicular
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precisely when this linear relation amongst the quadrances is satisfied q1 plus q2 equals q3
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the geometrical meaning is that you think about the quadrants here as the area of the square built on that side
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that's the way Euclid would have thought about pythagoras theorem it's a an equation about one
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area plus another area equals a third area that's the criterion for perpendicularity
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and then there's the triple quad formula which is sort of the sister theorem
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of pythagoras theorem which says that three points a1 a2 and a3 are collinear or in vector
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form a1 a3 is parallel to a2 a3 okay up here it was perpendicular here it's parallel precisely
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when this condition on the three quadrants is a satellite so q1 is the quadrant between a2 and a3
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q3 is the quadrants between a1 and a2 and q2 is the quadrant between a1 and a3 and the relation
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is that q1 plus q2 plus q3 squared equals 2 times q1 squared plus q2 squared plus q3 squared
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now that's considerably more complicated than than then pythagorean relation yes and that's why
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Euclid sort of didn't really find this although Archimedes would have have recognized this uh
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after giving it some thought so he was not far from understanding this
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okay so that's with quadrances now the crucial sort of new idea is how are you going to talk
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about the separation between two lines something that's more algebraic that you can actually
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compute that you can actually work with concretely and precisely and that's the notion of spread
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so you have two vectors in the plane say v and w the spread s between
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these vectors is defined as follows all right the spread between the vector v and the vector w
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is 1 minus this quotient v dot w squared on top quadrants of v quadrants of w
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on the bottom or in terms of dot products 1 minus v dot w squared over v dot v times w dot w
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in the Euclidean plane so when we're working with the the usual familiar dot product
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then this v dot w say the vectors are x 1 y 1 and x 2 y 2 then the dot product here is x 1 x 2 plus
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y 1 y 2 squared and the quadrances are x 1 squared plus y 1 squared x 2 squared plus y 2 squared and
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then there's a beautiful identity that goes back probably to Diophatus but certainly Fibonacci knew
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it that says you can rewrite this okay as x1 y2 minus x2 y1 squared over the same thing x1 squared
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plus y1 squared times x2 squared plus y2 squared this is a hugely important identity so if you have
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not seen it before please stop and verify that it actually is true by expanding appropriately
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now once we have the spread between two vectors well then we can extend that to a spread between
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lines because if you scale either these vectors v and w say you multiply v by a factor of 3 then
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this numerator here gets multiplied by a factor of 3 squared and the denominator also gets
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multiplied by a factor of 3 squared so the entire thing doesn't change so things are independent of
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scaling of the vectors involved so it really is a notion of spread between lines and here
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is a something of a spread protractor okay so if there's a horizontal line and then you have
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some other line okay so here might be a spread of 0.1 0.2 0.3 0.4 0.5 that's like 45 degrees
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up to a spread of one that's perpendicular and then it goes back down again symmetrically down
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to zero so from zero up to one but it's not a it's not a like an angle you see that these
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things are not sort of uniformly spread out it's it's different but it's um it's more beautiful
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now there's an important geometrical uh interpretation of the spread okay
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which you get from this second formula the one involving that identity of diaphones the formula
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that the spread between x1 y1 x2 y2 is x1 y2 minus x2y1 squared over the product of the quadrances
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so when we write it this way we we can recognize that what's in the numerator here is really just
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the determinant of the the matrix formed by these two vectors
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a squared and that's an interpretation of that's the like the area of that parallelogram
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squared so what we have here is a a way of thinking about the spread between um formed by two
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vectors that you're taking the parallelogram that they form and looking at the square of the area and normalizing that you know making it uh invariant under scaling right obviously if
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you increase v or w in the same direction well that parallelogram is going to get bigger by this sort of quadratic factors but if you put quadrants of v quadrants of w in the numerator
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then that's going to make that independent of scaling so another way of thinking about what
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the spread is it's just a normalized squared area of a parallelogram and that kind of makes
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it obvious that if the two vectors are parallel the spread is zero and then you know the spread is going to sort of increase up to a maximum of one and then it's going to decrease back to to zero
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now Paul Miller of the isle of wight who's one of my regular viewers and supporters some years ago
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came up with a remarkably different and simple way of calculating spreads which is quite beautiful
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so his uh spread protractor runs this way you you take a a semi-circle okay whose diameter
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is marked out by a ruler from zero to one okay just a linear scale and you're imagining that one
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of your lines that you're interested in measuring is horizontal along here and the other line is
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emanating from this rightmost endpoint on the circle okay so that's the blue line and we're
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interested in the spread between the horizontal and the blue line the spread s now on this uh
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semicircle you see currently a uniform scaling but in fact that uniform scaling is irrelevant okay that's not really necessary that's just to give you an idea so this blue line will meet the
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semicircle at some second point there's one point that it meets it'll meet at the second point and then you take that point of intersection and you just drop a perpendicular down to the base
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and then you get a number because the base is measured from zero to one in this case
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that might be something like 0.14 and that is the spread formed by the horizontal and this line here
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so notably this does not require a prior uniform subdivision of the circle okay it does not require
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that all it does is require the circle itself and being able to intersect this line with the
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circle and project down onto this linear scale so it's much simpler in fact to create such a thing
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than a traditional protractor very beautiful okay now there are aside from the pythagoras theorem
19:57
and the triple quad formula there are three additional laws of affine rational trigonometry
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i say i find rational trigonometry because there's also a projective rational trigonometry that's involved in hyperbolic geometry and spherical geometries and those formulas parallel the ones
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here but are more complicated so this is sort of the simple version we're just in the plane say
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the usual plane and we're talking about the laws of rational trigonometry so these laws replace
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um all the formulas basically that you learn in a usual trig course in the usual tree course you
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learn that there are three side lengths there are lengths and you learn there are three angles in
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fact in a general triangle you're not able to calculate any of these things quite honestly okay but nevertheless you learn that there's various relationships between them which are
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given by these transcendental functions functions which you actually also cannot compute exactly so
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everything is sort of working in some mysterious you know approximate computational universe here
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we're talking about precise computation where our computations actually run and they actually finish
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and you get an output that you can actually check okay here are the three laws they're the cross
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law the spread law and the triple spread formula and once you have these three laws then you have
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rational trigonology there's there's you know that that's it in terms of the fundamental formulas
21:28
there are many additional things that you can you know investigate but these are the basic laws
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so we have our triangle okay it has quadrances q1 q2 q3 so opposite the points a1 a2 and a3 and
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we have the spreads at each of the vertices s1 s2 s3 as indicated and here's my little symbol
21:49
for a quadrant a little rectangle along the line and the symbol for a spread is a line instead of
21:55
like a circular arc because there's nothing circular about this okay so we want to know some relationships between these six quantities cross law says that if we are interested in the
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three quadrances and one of the spreads say s3 although you know you could easily write down
22:14
the corresponding other two cross laws there's a relationship between the spread s3 and the three
22:21
quadrances and that is that q1 plus q2 minus q3 squared q3 because it sort of corresponds to s3
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equals 4 q1 q2 times 1 minus s3 so for example if you know the three quadrances then you have
22:41
essentially a linear equation to solve for s3 you might want to compare that with how easy it
22:48
is to find the angle of a triangle if you're in possession of the three sides in terms of lengths
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the spread law says that s1 over q1 equals s2 over q2 equals s3 over q3
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notice that there are no transcendental functions floating around it's just one number over another
23:14
number equals another number over another number etc and finally the triple spread formula which
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is the most complicated one is the relationship between the three spreads s1 s2 and s3 and it's
23:30
a lot like the triple quad formula except it has this additional cubic term so the formula is s1
23:38
plus s2 plus s3 squared equals 2 times s1 squared plus s2 squared plus s3 squared plus 4 s1 s2 s3
23:48
so all together a cubic relation amongst the three spreads formed by a triangle
23:57
this is in contrast with the the angular formula that says that if you have three
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angles say alpha one alpha two alpha three then alpha one plus alpha two plus alpha three equals
24:10
uh what is it three point one four one five nine two six uh and then it goes on for um um
24:20
forever okay that's that's the law that this is replacing this is just hugely simpler than
24:28
that law involving that infinite decimal pi that we don't actually really know about
24:39
okay there's a special case of the spread law here's the spread law again where you have a right
24:48
triangle say where the third spread here s3 is one that's that's that perpendicularity okay
24:56
then in that case if s3 equals 1 then you can solve for s1 and s2 then s1 is just q1 over q3
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and s2 is q2 over q3 so this is sort of telling us that in such a right triangle we can think
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about the spread s1 as being a ratio of quadrances like opposite quadrants over hypotenuse quadrants
25:22
so if you have learned some ordinary trigonometry this is like you know sine squared
25:30
okay of an angle but there's no signs it's just simple and then pythagoras theorem um well that
25:41
would be q1 plus q2 equals q3 so that has the form that that can sort of just be rewritten
25:46
that now as s1 plus s2 equals 1. so that's a property of a right triangle that this spread
25:54
plus this spread always add up to one okay that's essentially equivalent to pythagoras theorem
26:01
okay what i want to do now it will take uh just a little bit of time but it's i think really well worth doing is to actually show you proofs of these three laws okay so i want to impress
26:11
upon you that from a logical point of view this is all self-contained and that we don't have to
26:20
we don't have to invoke geometry and euclidean geometry or or some kind of calculus ideas so
26:27
if you actually think about the usual you know cosine laws and sine laws etc for for ordinary trigonometry the actual proofs of those are somewhat delicate
26:36
here it's all just basic linear algebra relying on the nice linear properties of the dot product that
26:44
we're starting with and we frame it in terms of the dot product making the the laws quite general
26:50
okay so let's look at the cross law okay here's the proof of the cross law okay it's not that hard
26:57
you really should go through it line by line listen very carefully to what i'm going to say
27:02
now okay so we're looking at this third quadrant's q3 okay it's the quadrants between a1 and a2 so
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it's the quadrants of the vector a1 a2 now the vector a1 a2 can be written as the vector a3a2
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that vector there minus a3 a1 okay this vector minus this vector is the same as that vector
27:26
so the quadrants of it is that vector dotted with itself now when you dot
27:33
this difference with this difference there's four things that happen you have a3 a2 dotted with itself that's just quadrants q1 okay because a3 a2 that's that vector there it's
27:47
dot product with itself is q1 and you also have a3 a1 dotted with a3 a1 with the altogether
27:54
plus sign that's going to give you a q2 and then you're going to get minus two times the
28:01
the other terms the a3 a2 dotted with a3 a1 there's gonna be two of them because
28:08
uh this thing dotted with this is the same as this thing dotted with this there's a minus sign
28:16
now what is this s3 so s3 is 1 minus the dot product of the vector a3a2 with a3a1 all squared
28:28
divided by the quadrances of those vectors which is uh q1 and q2 so it's one minus this dot product
28:36
but we've just figured out what this dot product is right um if we bring this to the other side
28:43
then we see that this dot product is q1 plus q2 minus q3 all over 2. so we can replace this
28:50
thing here with q1 plus q2 minus q3 all over 2 and when we square it we get this thing squared
28:56
and there's a 4 that appears down there and then you just clear denominators of this thing here
29:02
and you get q1 plus q2 minus q3 squared equals 4 q1 q2 times 1 minus s3
29:16
so all students of mathematics should go through this proof line by line making sure that they absolutely understand
29:25
that it works and it's completely general great okay now let's go to the spread law
29:32
okay so the same triangle we've just established this cross law right here it is right here
29:38
so we can just rewrite this as 4 q 1 q 2 s 3 just bring that to the other side and bring
29:44
all the other stuff to the other side so we have a 4 q 1 q 2 here and then minus this term here
29:49
q 1 plus q 2 minus q 3 squared and then it's an identity that this particular this difference here
29:56
can be written as q1 plus q2 plus q3 squared minus 2 times q1 squared plus q2 squared plus q3 squared
30:08
now that's uh something that you should recognize that's just the difference between the right
30:15
and left hand side of the triple quad formula so if the points are collinear then we know
30:22
that this quantity is zero so the non-zeroness of this quantity is some kind of reflection of how
30:30
non-collinear these points are so we actually define this quantity a as the quadrant of the triangle and it turns out to be 16 times the area of the square but that's on the side what's
30:39
important for our proof is to note that that that this quantity here is actually symmetrical in
30:45
q1 q2 and q3 it's symmetrical so although we calculated you know from the point of view of
30:52
s3 we're going to necessarily get the same thing if we calculate it from the point of view of s2 or s1 so we get this relation that this quantity 4 q1 q2 s3 equals 4 q1 q3 s2 equals 4 q2 q3 s1
31:08
and then if we divide everything here by 4 q1 q2 q3 then we get the relation that we want that s1
31:15
over q1 equals s2 over q2 equals s3 over q3 and as an added benefit we find out what that common
31:24
ratio value is it's the quadra over 4 times q1 q2 q3 again well worth spending
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half an hour if that's required going through it line by line making sure you understand everything
31:41
about this okay it's completely watertight we're not relying on any geometry off the page or some
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prior understanding of Euclidean geometry no okay and finally the triple spread formula the last
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of our three laws so we've just established the spread law 1 over q 1 equals s 2 over q 2 equals s
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3 over q 3. and let's write that as 1 over d so we know what that we could write out that what that
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d is but we don't really need to know what it is it's just some placeholder that we're going to use
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for the algebra okay so then we can re we replace these things here by writing q1 equals d times s1
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q2 equals d times s2 and q3 equals d times s3 and now when we sub this into the cross law
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there's the cross law okay so we replace q1 with d times s1 so there's a ds1 that
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appears here there's a ds2 that appears here there's a minus ds3 that appears here etcetera there's a ds1 that appears here there's a ds2 that appears here okay then all together on the
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left hand side there's going to be a d squared common factor and then s1 plus s2 minus s3 all squared on the right hand side there's going to be another d squared term coming from these
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things there's a four an s one s two times one minus s three and now the two d squareds on either
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side cancel and we're left with just a single relation just involving the s's s one s two s2 s3
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and that's a version of the triple spread formula but you can also just you know expand it and rewrite it and rewrite it in this
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more sort of obviously symmetrical form this is a this is a little bit longer but it makes it more
33:27
symmetrical so this is the triple spread formula as a consequence of the spread law
33:32
and the cross law and that ladies and gentlemen is is pretty well it those are proofs of the
33:40
the main laws of um rational trigonometry you might say well what about the um
33:46
the pythagoras theorem and the triple quad formula so it turns out that those are consequences of the
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cross law okay have a look here if you have the cross law and s3 happens to be equal to one that's
33:58
perpendicularity okay then that side's going to be zero so that's going to be zero that'll tell us that q1 plus q2 equals q3 and conversely if q1 plus q2 equals q3 that'll be zero so that
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s3 will have to be one so the pythagoras theorem is a consequence of the cross law and the um the
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triple quad formula is what happens when s3 is zero okay when s3 is zero well then this becomes
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four q1 q2 just times one okay so then you get a relationship between q1 q2 and q3 and that's just
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a an asymmetrical form of the of the um of the triple um quad formula okay so those first two uh
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results which i said are the two most important theorems in mathematics they are actually just consequences of the cross law special cases when s three equals zero and one
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so that's it it's a complete uh story um so that was uh probably about half an hour's
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worth that's probably a good place to stop um so in my next video i'm going to carry on
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and we're going to move towards understanding the spread polynomials in this lovely context.