Classical to Quantum | Towards rational rotations in physics | Wild Egg Maths
Classical to Quantum | Towards rational rotations in physics | Wild Egg Maths
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There is a big shift that is desperately needed in modern physics to bring it closer to having a reasonable chance at properly describing nature: by stepping back from the outdated usage of "angles" in many different contexts to describe rotations and coordinates on circles and spheres and even more generally in just coordinatizing motion. The alternative is to adopt more rational approaches: such as Rational Trigonometry which was introduced in my 2005 book "Divine Proportions: Rational Trigonometry to Universal Geometry". Another approach is "Vector Trigonometry" as described in the paper "Rotor Coordinates and Vector Trigonometry" available at https://www.heldermann-verlag.de/jgg/...In todays video we begin to introduce this more sensible and algebraic approach by expanding on our understanding of half-slopes as an alternative to angles regarding coordinates on the unit circle. Some very important formulas, essentially unknown in the literature, are here introduced, including the Triple half-slope formula and an analogous result for quadrilaterals. This is part of the Classical to Quantum series.
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0:03
hello everyone thanks for joining me on this adventure today we're going to be looking at rational rotations moving in this
0:11
direction so I have a very big task ahead of me of trying to get you physicists to let go a lot of your
0:20
common usage of angles in creating coordinate Frameworks and Analysis uh in
0:26
a wide variety of situations so it turns out that angles are not mathematically clearly defined objects
0:33
and contrary to popular understanding angles in fact angle measurements were
0:39
not actually involved with ID's original geometry in fact uid purposefully spent
0:46
a lot of effort avoiding talking about angle measurement so there are
0:52
alternatives to this so rational trigonometry is one alternative today I'm going to be talking about a variant
0:58
of that which I call vector trigonometry which is um ideally suited for
1:04
applications in the physical or engineering directions so what we're going to be
1:10
looking at more generally in the long term is trying to understand symmetries in physics and this is a very broad
1:16
topic but if we understand rational rotations if we understand just
1:21
rotations in a very simple algebraic way then this has the potential for
1:28
elaborating and explaining further and and giving us deeper insight into a
1:33
whole variety of things so symmetry in physics has essentially kind of two
1:39
aspects um the aspect of groups the aspect of hypergroups now hypergroups are much less well known and studied but
1:47
they are actually very important uh other side of the coin so with groups we
1:52
have many different groups but for example in the standard model we have S1 su2 su3 symmetry groups like orthogonal
1:59
groups and unitary groups we have Lorent po groups that appear in special general relativity then we have sort of fancier
2:07
extensions of these involved in things called Grand unified theories like su5 or s so10 or perhaps E6 E7 E8 and other
2:15
Alternatives so this is just a small segment of the groups that appear in in
2:21
physics but they definitely play a very key role on the hyper group side uh
2:26
hypergroup is roughly a group where the product of two things is a probability measure okay um which seems more
2:34
complicated than groups and some some level it is but it turns out that a lot of the hyper groups that appear are
2:40
actually commutative so the complexity of having to deal with probability measures is compensated for by the fact
2:47
that we're in a commutative framework so this roughly covers territory involving
2:52
group duels character tables so things that naturally sort of appear in the group theoretical situation have another
2:59
as aspect in fact a simplified aspect when you look at it from the hypergroup point of view things called tensor
3:05
categories West zinina Wht models fusion algebras and other things and in my
3:12
series defusion symmetry I'm going to attempt to try to lay out this side of
3:18
the Symmetry story okay so the challenge here is to
3:24
understand these various aspects of symmetry rationally or algebraically and
3:30
that means that we have to give up on the fictitious real number arithmetic
3:35
that currently we believe in artificially okay that doesn't really correspond to computational reality we
3:42
have to give up on algebraic closures like the algebraically closed field of complex numbers in which all your
3:48
algebraic dreams come true in just your ordinary irrational numbers like pi and
3:53
E and square root of two and cosine of 7 all those things okay so it turns out that we can go a long ways in
4:01
understanding these things without these fictions okay and the appealing U
4:09
Prospect here is that by doing that by clearing the board of these things which don't really work logically we have the
4:16
potential of understanding what's really happening in a more fundamental and crucial way so we can make progress
4:26
right we can make progress in terms of our understanding by clearing out the underlying mathematics
4:32
of these things which are basically you know religious positions does that sound appealing to a
4:38
physicist it should right it should sound appealing because what I'm really proposing is that we move closer to a
4:45
mathematics which is grounded in computational reality for which there's a theoretical result and over here we
4:51
can practically demonstrate that in its entirety on our computer that's a powerful mathematics that has the
4:58
potential of renovating the physical landscape the underlying mathematics of
5:03
the physical landscape so very very important so the starting point for us will be to understand
5:10
rotations starting with rotations in two dimensions and then in three dimensions and then in four dimensions and then
5:18
more complicated things we're going to think about orthogonal and unitary groups and higher Dimensions uh Spinners Clifford algebras
5:25
or grason Clifford henus algebra sometimes called um geometric algebra
5:32
which is a name I don't like but uh these are all um things that we can try
5:38
to get hold of rationally starting with an understanding of rotations in low
5:43
Dimensions by the way I guarantee that if you're a working physicist by the end of this video okay you will appreciate
5:50
that there is a dramatically different way of thinking about rotations in the plane okay and the the possibility will
5:59
emerge for you that all those spherical coordinates and cylindrical coordinates and various angles that you routinely
6:07
introduce into physical theories are perhaps replaceable and that the resulting
6:14
mathematics is going to be more algebraic more rational and more powerful okay after we' sort of get our
6:22
heads around this kind of stuff then we can move to somewhat more advanced theories representation Theory or
6:27
harmonic analysis for transform theory for simple compact leag groups and the
6:34
simple compact leag groups um you know they're these sort of four families a and DN BN CN families corresponding to
6:42
families of um well General linear groups uh orthogonal groups and slec
6:49
groups and then further on there's these exceptional groups G2 F4 E6 E7 E8 which
6:58
play a strange role are connected with all kinds of things but they're also happening in this family although
7:04
they're more complicated and then beyond that sort of further down the road we have apine Lee algebras more generally
7:10
cats Moody infinite dimensional Le groups and algebras so you know this whole story gets sort of ramped up to
7:16
infinite dimensions in some fashion and another direction is to uh move from
7:22
compact which are sort of you know bounded objects to to things which which go on beyond our view so the theory of
7:28
non-compact groups for which the harmonic analysis is more complicated and infinite dimensional versions of
7:34
them and there's categorical Theory aspects of this Quantum groups and and whole bunch of other interesting
7:40
mathematical uh topics get woven into the story okay so there's a possibility
7:45
of really thinking differently about a whole wide range of mathematical things
7:51
that have very important consequences for physics okay so the crucial starting
7:56
point is in two Dimensions so we have two dimens space for us that's typically
8:01
over the rational numbers although we have sort of our gaze also to finite Fields what would happen if we worked
8:08
over a finite field a prime field say but the default is over the rational
8:14
numbers okay X Y axis axes like this here is a unit circle as usual and um
8:21
here's a vector okay we're going to think about these is a vector space okay so Vector may be given by round brackets
8:28
XY a row Vector for here's another vector v Prime X Prime y Prime and these two vectors are such
8:34
that their extent from the origin is the same so we can U think about a rotation
8:40
of the plane which fixes the uh origin and moves V to V Prime let's call that
8:46
transformation row so what we're looking for here is a linear transformation in this Vector space linear transformation
8:53
so it preserves the origin and it crucially preserves the dot product the
8:58
dot product is the linear algebra way we impose the metrical structure okay and
9:05
here is the standard dot product that if say we take these two vectors their dot product is x * X prime plus y * y Prime
9:13
and our linear transformation the rotation that we're considering should preserve this dot product in the sense
9:19
that row of V V and V Prime are any two vectors row of V do row of V Prime
9:25
should be the same as v. V Prime preserves the dot product now it's also possible to restate this in terms of an
9:31
Associated quadratic form Q so Q of the vector v is in this case that V is XY is
9:38
the number x2 + y2 okay and when we say linear
9:44
transformation it means we're talking about matrices 2 x two matrices in this case we're looking at row vectors so the
9:51
matrices act on the row vectors on the right so if the Matrix is a b c d then I hope we all realize that the product of
9:58
this Vector with this Matrix is ax + c y That's the first term and BX + Dy that's
10:04
the second term all right so that's we're looking for rotation of the plane of this form
10:10
depending on four numbers okay and um yeah and and we want to study these
10:18
rotations okay now um in terms of linear algebra since
10:25
we're working with this dot product it's convenient to um recognize that we can write that in Matrix kind of language so
10:33
if V is XY and V Prime is X Prime y Prime in general okay then this dot
10:39
product here this uh this number x x prime plus y y Prime can also be thought
10:44
of as the product of the row vector v and the column vector v Prime transpose
10:50
so you write the row Vector XY and you write the column Vector X Prime y Prime you do the ordinary matrix
10:56
multiplication x * X prime plus y y Prime and you getting this dot product
11:01
so we can translate this condition if we like purely into the language of linear
11:06
algebra okay so that's the usual setup so in the standard story you are
11:13
going to be talking about angles you can say ah rotation like this is determined by an angle Theta and then you're going
11:19
to express the corresponding Matrix in terms of cosine Theta and sin thetas
11:25
okay but you have to acknowledge that the the angle Theta that you're talking
11:30
about typically is an arc length that's perhaps the most usual way to think about it's actually the arc length of
11:36
this little circular Arc between here and here and that requires calculus to
11:41
Define and if you look at the calculus it requires infinite calculation that is
11:48
unbounded calculation there's algorithms that you have to carry on to Infinity before you typically get angles that
11:55
means that although even though these vectors here may be very concrete okay like the vector 34 and the vector
12:01
43 or something like that writing down what the angle between them is exactly
12:08
is something that's computationally beyond Us in general okay so we can't
12:14
actually honestly compute these things and that means from a pure Mass point of view okay that they're not actually
12:21
really properly Define mathematical objects so we should be looking to
12:27
having an alternate Theory where we can describe rotations without angles
12:34
physicist this is so important to get your head around I know this is hard to to swallow that that there might be such
12:41
a thing and that it might actually make your lives a lot easier ultimately okay
12:46
but it that's the way it is okay um physics where you replace angles most of
12:53
the time with rational analoges is going to be a much better physics
12:59
okay so now let me present the usual rational rotation story something that I've explained in dozens of locations
13:06
probably throughout my videos but we're going to go further today okay so even if you have watched these things and
13:12
seen this formula many times you're going to learn more things some really important additional things here today
13:19
under the general umbrella of vector trigonometry okay so the starting point
13:25
is that if you have a vector which is a unit vector Vector this is like this V
13:31
so starting from the origin and emanating to a point on the unit circle
13:36
then that Vector can be described rationally in terms of a parameter T and
13:43
the geometrical significance of T is is that it's a point on the Y AIS say right
13:48
there and we're connecting this point minus one on the x- axis to this point T and find the second point of
13:54
intersection of this line and it's a little computation that tells us that this point called a e of T is 1 - t2
14:02
over 1 + t2 comma 2 T over 1 + t2 so as a vector that's the vector v okay so
14:09
first of all that
14:28
coordinatization this terminology actually was suggested to me by my colleague Daniel Chan some years ago
14:34
okay it's quite a nice terminology so the half slope of the vector v is this number H so in this case um what would
14:43
the half slope be well this point t on the y- axis looks like it might be uh
14:49
you know 37 okay um so that would be the half
14:56
slope of this particular vector v and we're going to use this this coordinate
15:03
here to define a rotation given by a 2 X2 Matrix so we're going to think about
15:08
the rotation that takes the vector from 0 to one so our standard basis vector
15:14
and rotates everything up to the vector v okay so this rotation we're going to
15:20
call it Rose of H because it depends on T or H um we're going to write this
15:25
thing here in this Matrix form and just to make things a little bit simpler I'm taking the common denominators out so we
15:32
have 1 over 1 + h2 and then we have 1 - h 2 2 H - 2H and 1 - h2 so this is a
15:40
an orthogonal Matrix okay it has actually determinant one okay it
15:46
satisfies the condition that if you call this Matrix Q then Q * Q transpose is
15:51
equal to the 2x2 identity Matrix and this is then as a way of describing
15:56
rotations in the plane so this is a rational alternative to the usual cosine
16:02
Theta sin Theta minus sin Theta cosine Theta that you find typically in physics courses and in texts and papers okay so
16:11
if we're going to adopt this rational point of view the one of the first orders of business is to say okay what
16:17
does the group structure look like if we have two rotations say row of H1 and row
16:23
of H2 then the product of those two matrices will be another rotation of the
16:29
same kind but the parameter will be something different H3 so how do you find H3 in terms of H1 and H2 it's given
16:36
by this formula okay we introduce What's called the circle sum so H1 Circle sum H2 by definition
16:45
this is going to be H1 + H2 / 1 minus the product H1 *
16:53
H2 now as a number obviously there's a little bit of problem what happens if H1
17:00
* H2 happens to be equal to 1 well in that case at this stage here this is not
17:06
defined although we're soon going to see a remedy for that okay so this is a nice
17:12
little computation so you just take this Matrix here and you calculate that and you check that this particular thing
17:18
works and that uh the matrix multiplication is a suggested by this formula here with this value of
17:26
H3 okay once we've done done that introduce this circle sum then we can
17:32
introduce the relative half slope between two unit vectors so suppose we have one unit Vector say V1 and another
17:40
unit Vector say V2 and we want to talk about the relative half slope between
17:45
them which is to say we're looking for the rotation that sends the V1 Vector to
17:51
the V2 Vector okay so this one here was designed to talk about the the rotation
17:57
that sends the standard basis Vector to an arbitrary vector v now we're being a bit more General how do we rotate from
18:04
the vector V1 to another Vector V2 of course it doesn't have to be over here it could be over here also okay so we're
18:10
going to introduce the half slope relative to these two vectors so the relative half slope h of V1 V2 well that
18:18
turns out to be H2 Circle sum minus H1 okay uh plug it in you get H2 minus
18:26
H1 / 1 + H1 H2 okay so that's a way of calculating
18:35
then the rotation required to go from one unit Vector to another unit Vector
18:41
once you have the half slopes of the two unit vectors in hand it's exciting isn't it so I'm going
18:49
to present a whole number of formulas uh today interesting and important results
18:54
and I'm hoping that you physics minded people will check these formulas so it's no big deal to sit down for a
19:02
little while and verify algebraically that the various things that I'm saying here are true and to further uh convince
19:09
yourself of them by working through different examples all right so that's important to do okay so our first term
19:17
is a half slope formula which is a nice compact formula that tells us how to get
19:24
the half slope of a vector uh just by a very simple computation so if V equals
19:31
XY is a unit Vector so for example perhaps that one there okay there's vector v from the origin to that point
19:38
there say the vector X Y then the half slope okay that's going to be the
19:44
coordinate on the Y AIS H there so that this line from minus1 through H meets
19:50
this point on the unit circle okay and statement is that this H depends on V is
19:57
y divided by 1 + x that's all you take the y-coordinate
20:04
and you divide it by 1 + the x coordinate and you have the half
20:11
slope of course there's a provise also when I write down a formula like this you should look at the denominator and
20:17
say oh well maybe the denominator could be zero so the formula is going to be automatically sort of excluded when the
20:23
denominator is zero so in this case that's when X is minus one so if x is minus one well that's only going to
20:29
happen actually when we're at this point here it's the only place where that formula is not going to work
20:36
okay all right so let's do an example so suppose that V is the vector
20:43
513 minus 12 13 so that could be that point there okay that's that Vector
20:50
there of course that's connected with a Pythagorean triple at 52 + 12 = 13 s
20:56
that's why we have such a nice r point on the circle of that form because there's a Pythagorean
21:02
triple under the Bonnet here okay so what is the half slope for this thing
21:08
well we take Y which is -12 over 13 and we divide by 1 + x 1 + 5 over 13 we
21:16
multiply through by the common denominator 13 we get -2 over 18 which
21:22
simplifies to - 2/3 so the half slope of this Vector here is - 2/3
21:29
and now physicist please I would like you to seriously think okay suppose I asked you okay what's the corresponding
21:36
angle right traditional physics you're going to describe that same Vector there's the vector okay you going
21:43
describe it by an angle so please tell me what is that angle and I'm happy you
21:48
can tell me in degrees or in radians either one well you can guess but I want you to
21:56
compute that angle and of course you know right I hope you really appreciate that you can't calculate that angle even
22:03
if I paid you a billion dollars you cannot calculate that angle you can only
22:09
obtain some approximate formula for it or you know or write some expression
22:15
like tan inverse something or other or ar cosine something or other okay which is just a short form for an algorithm
22:23
which you can't actually run to its end are we all agreed on that that's so
22:29
important to acknowledge that the angle is something which is in in fact not
22:36
properly defined it's not just that you can't compute it it's that it is not a
22:42
properly defined mathematical object the half slope has no such
22:48
problem it's just there so to extend this a little bit and
22:56
to sort of get at this sort of special point and the corresponding rotation uh
23:02
we could introduce e of infinity right after all if you take the formula and you for T and you let T go to either
23:09
positive infinity or minus infinity you're going to see that the the vector is approaching this point minus one 0
23:16
here so we could Define e of infinity to be minus one 0 and the corresponding rotation will be minus one
23:23
0 0 minus one which is just uh if you like reflection in the origin or a
23:29
half a turn okay but that does sort of imply
23:35
that we're sort of extending our number system here we have the rationals you know and now we're including one more
23:41
point at Infinity so you know there's a question about um to what extent can we
23:46
do arithmetic consistently in this sort of bigger domain that's a question I've talked a lot about in my math Foundation
23:53
series and it's a it's a very relevant question here okay now um a very
23:58
important formula which very likely you haven't seen before it's what I call the triple half slope
24:06
formula okay and it says that if V1 V2 and V3 are unit vectors and here are the
24:14
three relative half slopes between them so h12 is H of V1 V2 h23 is H V2 V3 and
24:24
h31 is H V3 V1 so these are the three sort of relative half slopes between
24:31
these three vectors taken in the order V1 V2 V3 and then back to V1 then these
24:38
three half slopes satisfy this relation not too hard to remember saying
24:45
that their sum is equal to their product h12 plus h23 plus h31 equals h12 * h23 *
24:56
h31 now you have to admit that's a pretty beautiful formula okay that's a pretty
25:03
beautiful formula you might say yeah but it's not quite as uh pretty as what happens with
25:10
angles with angles you have something like Theta 1 plus Theta 2 plus theta 3 equals zero that's
25:16
nicer yeah maybe if you don't take into consideration that that the Theta 1
25:23
Theta 2 and theta 3 each of them require an infinite amount of computation to get hold of and that that equality that
25:30
we've just talked about only holds once you have all three of them to infinite
25:35
Precision which takes you an infinite amount of time times three
25:40
okay these half slopes are not like that at all you calculate them in a relatively small amount of time so the
25:48
formula is just a little bit a little bit more complicated but at the advantage that you're talking about
25:54
things which actually do make precise exact sense
25:59
okay let's look at an example here is V1 it's e of minus 23 what's that mean well
26:07
here is minus 2/3 on the Y AIS We join that there that's that point there so
26:13
that's the vector V1 and uh we calculate just using the formula that it's 5 13 -
26:20
12 over 13 we've seen that point that Vector before okay V2 is e of one qu so
26:26
there is one quarter okay join that up we get that point there plug into the formula you get 5 over 17 8 over 17
26:34
corresponding to the Pythagorean triple 82 + 152 = 172 and V3 this Vector here is e of two
26:45
so on the y- axis we look at the point whose coordinate is two We join that to
26:51
minus one there that line and it meets the unit circle at that point so that Vector there from the origin to that
26:57
point is V V3 that's e of 2 so the half slopes of V1 V2 and V3 are- 2/3 1/4 and
27:09
two respectively so this would be H1 this is H2 this is
27:15
H3 okay so now we can talk about the um the relative half slopes so let's look
27:21
at h12 that's the the half slope between V1 and V2 and I should maybe
27:28
tell you how I like to signify that so we're going from V1 to V2 so it is sort of directed so I like to put a little
27:34
arrow like that okay it's not like a spread a spread doesn't have an arrow this has an arrow so I would put maybe
27:41
h12 beside that that's a two okay okay how do we calculate h12 well it's H2
27:48
Circle sum minus H1 so that was H2 minus
27:54
H1 so one qu or minus minus 2/3 divided 1 + H1 * H2 okay there's H2 there's H1
28:05
so 1/4 + 2/3 / 1 minus uh 1/4 * 23 and
28:15
you calculate that and you get 11/ 10 to get h23 we look at the half slopes
28:22
H2 and H3 so it's H3 - H2 / 1 + h h 2 *
28:29
H3 that turns out to be 76 and h31 going from here to here okay
28:36
that's H1 minus H3 / 1+ the product H1 *
28:42
H3 that turns out to be 8 so let's let's just uh talk about that
28:47
okay so what does it mean for h12 to be 11 over 10 that's an expression of how
28:53
far those two vectors are apart now there pretty close to being
28:59
perpendicular but not quite so what does this mean that this half slope is 11
29:06
over 10 well it means that this thing here has sort of the same separation as what we would get if we rotated this
29:12
thing so that this first Vector would be the standard basis Vector going from 0o to one so if we rotated this okay like
29:19
that then this Vector here would be rotated to some point somewhere over there and so this relative half slope is
29:25
really the same as the the half slope of that point which we get by joining this up to the minus one point and seeing
29:32
where it meets the vertical axis and what we're saying that would be 11 over
29:37
10 11 over 10 that's the line there and then that would be the relative path
29:44
slope you might like to use dot products and and inverse cosiness or inverse Tans
29:50
or whatever you like in order to calculate the angle between those two vectors good luck because you're not
29:56
going to be able to actually make that calculation correctly okay that's what I mean I'm a
30:03
pure mathematician when I say make a calculation I mean correctly completely
30:09
precisely the answer is on the page everyone can check that it's 100%
30:15
correct it's a mental deficiency that us pure mathematicians have we like to be completely exact when we
30:22
can okay and then there's a corresponding h23 what's the meaning of that between here and here so if we
30:29
rotated this pair so that uh the the first one is at the origin would be sort over there and then we're saying that
30:35
the half slope is 76 so we have to go a little bit higher and we go there and that would be also in that direction and
30:42
this last one is eight what does that mean that means that's between V1 and V3
30:47
so um but actually from from V3 to to V1 so if we rotate this thing around okay
30:54
so that V3 is in this direction then V1 would be sort of over there somewhere
31:00
what would its half slope be well we would see where it meets we would draw a line between that and the point minus
31:06
one and we would see where that line meets the vertical axis and we're saying that's it's going to meet it at the
31:11
point with coordinate 8 okay so very specific thing we have
31:17
three unit vectors we've calculated their three relative half slopes these
31:24
numbers 11 over 10 7 over 6 and 8 and now we can verify by the formula the sum
31:30
of these three things is 154 over 15 and the product of those things is also 154
31:37
over 15 let me make another observation
31:43
before we leave this that this formula is actually relatively simple if
31:49
you have two of them so if we knew h12 and h23 and we wanted to find h31 well it's a linear equation in h31
31:58
so we can solve that immediately okay so two more results uh here Circle sum associativity the circle
32:06
sum of two numbers so if H1 and H2 numbers H1 Circle sum H2 I remind you
32:12
that's H1 plus H2 in the numerator and divided by 1us H1 * H2 in the
32:20
denominator so that's associative in the sense that H1 Circle sum H2 Circle sum
32:25
H3 is equal to H1 Circle sum H2 Circle Su H3 so we can just dispense with the
32:32
brackets and just write H1 Circle Su H2 Circle sum H3 now if we remind ourselves that this
32:41
circle sum corresponds to composition of corresponding rotations then this is pretty believable
32:48
because we know that rotations are associative okay but if we just had the
32:55
formula this would be something to check and the theorem says that actually what
33:01
you actually get is explicitly this okay you get H1 plus H2 plus H3 minus H1 H2
33:08
H3 I notice that's the essentially the numerator is what appears in that
33:14
previous formula all right that I just derived and in the denominator we have 1
33:20
minus H1 H2 plus H2 H3 plus H1 H3 so we
33:25
have one we have sort of the elementary linear symmetric function of H1 H2 H3
33:31
here we have sort of the elementary quadratic symmetric function of H1 H2 A3 and here we have uh the elementary cubic
33:39
symmetric function of H1 H2 H3 so um
33:45
nice exercise to prove this and to generalize it generalizing means the
33:50
obvious thing what happens if you take a circle sum say of four or more numbers
33:58
okay now the quadruple half slope formula now suppose that we have four
34:03
unit vectors V1 V2 V3 V4 unit vectors
34:09
and we Define relative half slopes h12 which is H of V1 V2 correspondingly h23
34:16
between V2 and V3 h34 between V3 and V4 and h41 coming back between V4 and V1 so
34:24
now we have four half slopes well well then here's the relation that the sum
34:31
h12 plus H2 3 plus h34 plus h41 is equal to this sum of four cubic terms so here
34:40
is the the product of three of them here's another product of three of them
34:45
here's another product of three of them here's another product of three of them so we're looking at all four products of three of thems um in in and adding them
34:53
all up um again this is a linear equation in
34:59
any one of them so if we knew three of them say we didn't know h41 it appears there it appears there linearly it
35:05
appears there and there linearly so we get a linear equation in h41 as well so this sort of trigonometry with half
35:12
slopes instead of angles does that appear in my book Divine proportions no
35:18
it doesn't because it's not actually rational trigonometry it's almost rational trigonometry but it isn't quite
35:25
it's um a trigonometry which is sometimes rational but it has some irrational or unbounded or approximate
35:32
aspects to it so I want to just in my last slide here uh talk about this okay
35:37
because it also uh explains there's an important formula here that shows you
35:43
how to extend these Notions to more General vectors not just unit vectors
35:49
okay so here's the the plane as usual here's our unit circle uh now I'm considering uh three vectors which are
35:57
not unit vectors so the first Vector V1 is the vector 1 minus one 1 - one from
36:03
here to here the second Vector V2 is the vector 21 2 1 and the third Vector is
36:10
the vector -1 -3 that Vector there okay so first of
36:18
all how do you define the half slope can you define the half slope of one of these vectors and the answer is yes but
36:25
in general only approxim imely so for a unit Vector the half slope is completely
36:32
defined and is a rational number here it's not the case generally so let's look at V V1
36:40
here okay what is the half slope of this thing well you might say well what we
36:45
should do is we should just look at the corresponding unit Vector so we should scale the vector by dividing by its
36:53
length and of course that's something that comes very naturally to physicists but I want you to be aware that the the
37:02
size the length the distance uh quantity involves a square root okay so that
37:08
square root is in general a procedure that requires an infinite amount of work
37:14
so whenever you take a square root you are leaving the rational World okay um
37:21
and so it's still useful okay but you are then definitely in an approximate or
37:27
applied math setup so we just want to acknowledge that in particular from a pure math point of
37:34
view there is no actual point on the unit circle which lies on this line uh x
37:40
= minus y okay because we don't have a root2 in the rational world so there's
37:46
no no such Point although we can find approximate rational points so what
37:51
we're going to do is we're going to um you know um frame things now in terms of
37:58
approximate things so the The crucial thing is to look at say R which is the
38:03
the length of V1 okay which is obtained from the quadrants by taking the square
38:09
root of the quadrant so square root of V1 V1 and we recognize that this is an approximate thing so I put little quotes
38:15
above here right so you're not supposed to think of this as being something exact this is something approximate
38:20
you've got some square root calculation you have to get your calculator or or work it by hand up to a certain point and then you truncate and then you have
38:27
some approximate thing but it's not the real square root it's just an approximate value but let's call that R okay so
38:34
here's the formula for the half slope of the vector V1 it's take the X and Y
38:39
coordinates it's y/ r + x very similar to Y over 1 + x okay that
38:47
was the formula for the unit Vector here it's y/ r + x where R is the perhaps
38:54
approximate length of the vector V1 so this half slope will be an
38:59
approximate thing and in this case here well the approximate length is < tk2 so
39:05
we get U -1 over < tk2 plus the x coordinate is 1 and that's approximately
39:12
- 414 so that's still giving us information right it's saying that if we
39:17
want to go in this direction the point on the unit circle or the a point that's pretty close to the unit circle is
39:23
obtained by going to the point minus 414 which is right there and joining that up
39:28
there and then where that line meets the circle that would be an approximation to the corresponding U point that we
39:35
require the half slope H2 where is uh here's V2 okay so has approximate length
39:42
Square < TK of 5 so the same formula 1/ < TK 5 + 2 gives a approximate half
39:48
slope of. 236 meaning that that point there would have coordinate 236 and H3
39:55
which is um the half slope for this Vector V3 okay
40:00
minus 3 is the y-coordinate and the the r isqu < TK of 10 and the x coordinate
40:07
is -1 so we get -3 over < TK of 10 - 1 which is approximately -
40:14
1387 okay so we have now approximate values but that's that's the way we do
40:20
um trigonometry usually okay you're used to that but the nature of the
40:25
approximations this way is a little bit simpler in fact it's a lot simpler than the ones that we usually use for
40:30
trigonometry because you're not involving these cosiness and SS and tans and Arc Tans and so on so
40:37
algebraically um a square root of two is a much more tractable object than something like cosine of 7 or or tan
40:45
inverse of 739 okay and one way of of
40:51
seeing that is first of all the algorithm for calculating it is is demonstrably simpler than algorithms for
40:56
the other things um but also from another point of view you know if we really wanted to make this rational we
41:03
could U make an extension of our field okay so we could extend our field to
41:09
include say an algebraic route too and if we wanted to talk about all of these things we could also you know extend
41:16
further to get an algebraic square root of five and then we would be in the possession of being able to talk algebraically about all of these things
41:23
so algebraic extensions finite extension Fields typically with Matrix es would
41:28
allow you to actually do all of this arithmetically completely exact but it's
41:33
going to be different from this decimal approximation business that I'm giving you here okay so let's let's just stick
41:40
with the decimals here and calculate a a relative half slope say h12 between
41:46
Vector V1 and V2 so this half slope h12 well it's H2 Circle sum minus H1 so
41:53
that's H2 minus H1 over 1 plus H1 H2 there's H1 there's H2 we um calculate
41:59
that say to three decimal places because that's what we're using we get 720 h23
42:06
between this vector and this Vector is uh turns out to be minus 2.41 3 and
42:13
831 between this vector and this Vector
42:18
uh turns out to be 61833 988 and so on
42:27
and you might ask why am I giving you uh more digits nine decimal digits instead of just three well because you might
42:34
recognize that this is a sort of an important decimal okay um this is
42:41
essentially a golden ratio okay um so there's something sort of magical about
42:48
this particular configuration between this vector and this Vector in that the
42:53
the half slope that's determined there actually has an intimate connection with
42:58
golden ratio so that's something that maybe doesn't appear so so readily in ordinary
43:07
U geometry but it's sort of interesting but in any case I just want to verify the uh the triple half slope formula so
43:17
if we take h12 plus H2 3 plus h31 with these values we get something approximately minus
43:24
1.75 and the product of the three of them is is approximately minus 1.74 so you know we're still getting the
43:31
same laws appearing here uh but we have to work in an approximate setting okay
43:38
so this whole story of vector trigonometry is U well I've written one
43:43
paper and I have a few preprints uh on it so I I probably should say a lot more about this at some point but already
43:50
here we see that this crucial notion of a half slope this is the main idea I want to get through to you that the
43:56
notion of a half slope say between two vectors is a very viable
44:03
alternative to uh to angles and it allows us to get
44:09
very good control over rotations okay so we're just talking about rotations that we're always going
44:15
from unit vectors to unit vectors so the the rational story Works to describe
44:20
rotations completely okay and we get beautiful sort of alternate formulas to
44:27
deal with um the the S SO2 rotation Group which is admittedly very simple
44:33
but the possibility now emerges that maybe we can go to three dimensions and
44:38
replace spherical coordinates and such with rational analoges and maybe we can
44:44
go to higher dimensions and start thinking about rotations and orthogonal
44:50
matrices and unitary matrices and maybe these other Le groups with a more
44:55
rational orientation
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