Advice for Maths
Is there a need for a dedicated two-dimensional version of Neil Sloane's Online Encyclopedia of Integer Sequences? Now the OEIS does deal with such two-dimensional arrays, but somewhat incidentally. I think it would be useful to have an additional such resource that concentrates on the two-dimensional case, and in particular highlights the rich aspect that the linear algebra of ongoing matrices, or on-maxels, introduces.
Let's discuss these ideas in the explicit context of the Pascal- Harriot array / maxel and investigate some secondary sequences that can be easily extracted from it. This array is in my opinion amongst the most important objects in the entire mathematical universe! It is therefore well worth careful study from multiple points of view.
The techniques we discuss in this lecture are rather simple minded, but I hope you can see that they have very wide application.
Here are the Wild Egg Maths Playlists (some available only to Members!)
Algebraic Calculus One (for the Open Learning Course) : https://www.youtube.com/playlist?list...
Algebraic Calculus Two (for the Open Learning Course in preparation): https://www.youtube.com/playlist?list...
Advice for Mathematics Exploration (for Members only): https://www.youtube.com/playlist?list...
Solving Polynomial Equations (Members only): https://www.youtube.com/playlist?list...
Hexagrammum Mysticum (Members only): https://www.youtube.com/playlist?list...
Algebraic Calculus and de Casteljau Bezier curves (Members only): https://www.youtube.com/playlist?list...
Exceptional Structures in Maths and Physics via Dynamics on Graphs (Members only): https://www.youtube.com/playlist?list...
Maxel Inverses and Orthogonal Polynomials (for Members only): https://www.youtube.com/playlist?list...
Triangle Centres: https://www.youtube.com/watch?v=iLBGX...
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...
*
Here are the Insights into Mathematics Playlists:
Maths and Music: https://www.youtube.com/playlist?list...
Diffusion Symmetry: https://www.youtube.com/playlist?list...
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Playing Go: https://www.youtube.com/playlist?list...
Year 9 Maths: https://www.youtube.com/playlist?list...
Ancient Mathematics: https://www.youtube.com/playlist?list...
Wild West Banking: https://www.youtube.com/playlist?list...
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list...
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list...
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list...
MathFoundations: https://www.youtube.com/playlist?list...
Wild Linear Algebra: https://www.youtube.com/playlist?list...
Famous Math Problems: https://www.youtube.com/playlist?list...
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...
Differential Geometry: https://www.youtube.com/playlist?list...
Algebraic Topology: https://www.youtube.com/playlist?list...
Math Seminars: https://www.youtube.com/playlist?list...
0:01
hello everyone i'm norman wahlberg
0:02
thanks for joining me on this adventure
0:04
we're looking at maxwell theory number
0:06
theory in a bigger way in a
0:07
two-dimensional kind of fashion and i'm
0:09
illustrating things with families of
0:11
famous polynomials or polynomials
0:14
i'm trying to convince you that looking
0:16
at them from the linear algebra point of
0:17
view is profitable and important
0:20
so when we move in this direction i
0:23
think a natural thing that starts to
0:24
occur to us is that the wonderful online
0:27
encyclopedia of integer sequences that's
0:30
neil sloane's 1964 creation
0:33
which has now been augmented by
0:35
thousands of contributions from wide
0:37
range of people
0:38
that perhaps there's something else
0:42
looming ahead which is in the rough
0:45
framework of a two-dimensional version
0:48
of the online encyclopedia
0:50
and i think it's important to start
0:52
having a discussion about what
0:54
that might might look like and how one
0:57
might go about doing that i think
0:58
there's interesting questions uh to be
1:00
raised there so to to make the uh
1:03
discussion rather specific i want to
1:06
sort of kill two birds with one stone
1:08
and look at the pascal harriet maxill
1:10
again
1:11
this is such an important key object in
1:13
mathematics and
1:15
it'll be interesting for us to to see
1:17
how different structures can be
1:20
extracted from it and how those
1:23
other structures can then create new
1:26
structures in addition
1:28
so there's some sense here that the
1:30
two-dimensional situation is perhaps
1:32
more
1:34
rigid or a little bit more structured
1:36
more amenable to classification than the
1:38
one-dimensional analysis of sequences
1:41
this is maybe very roughly parallel to
1:44
the the distinction between sort of
1:46
one-dimensional a real
1:49
differential analysis and
1:51
one-dimensional complex
1:54
analysis or perhaps the distinction
1:56
between c infinity and analytic
1:58
categories very roughly
2:00
all right so let's start with the
2:02
fundamental pascal harriet maxwell
2:05
which can be defined very simply the ij
2:09
entry is just this binomial coefficient
2:11
i choose j
2:13
now
2:15
our indexing is is starting at zero
2:17
that's the usual convention now it might
2:19
actually be preferable for some
2:20
applications to use the notation that
2:23
the physicists like and to put one of
2:25
these indices as a subscript or the
2:26
other one as a superscript say p sub i
2:29
super j
2:30
so that makes it clear that we can
2:32
extract two kinds of one-dimensional
2:35
sequences from this two-dimensional
2:37
array so we can let p i p sub i then be
2:41
the finite sequence given by the i throw
2:44
in other words p sub i of j is p
2:48
i j
2:50
okay so if you fix i then you get a
2:52
function of j that's a that's a sequence
2:55
which ends up reading off one of these
2:57
rows uh let me just scroll down here a
2:59
little bit so there's
3:02
the pascal
3:03
harriet maxwell
3:05
and so for example
3:09
so for example what would p
3:13
p
3:14
sub 3
3:15
would be
3:17
the sequence
3:19
one
3:20
three three one
3:22
and correspondingly we could let p super
3:24
j be the on sequence of the j column so
3:28
looking down a column well then we don't
3:30
get a finite sequence anymore we get one
3:32
of these
3:33
so on sequences that just keep on going
3:36
so for example and let's look at say p
3:38
sub
3:39
4
3:40
well that's going to be
3:42
0 1 2 3 4. so that's going to be the
3:45
sequence that starts 1
3:47
5
3:49
15
3:50
35
3:52
etc c dots
3:54
okay that's an on sequence
3:56
so
3:58
we obviously have the rich family of
4:01
these row sequences and column sequences
4:04
associated with such
4:06
a two-dimensional array
4:10
let's
4:12
also just observe that however we can
4:14
create other kinds of
4:16
sequences typically from a
4:17
two-dimensional array because we can
4:19
read this thing in a number of different
4:21
ways so another obvious kind of thing to
4:23
do is to read down the diagonal okay if
4:25
we read down the diagonal
4:27
well we get one one one we actually get
4:28
the same sequences in the first column
4:30
and one of the beautiful things about
4:32
this pascal area array
4:35
is that when you do read down the sub
4:37
diagonals so the next one would be one
4:39
two three four five six that's
4:40
essentially the same as the
4:42
the next column
4:44
except that the next column starts with
4:45
a zero but other than that uh the
4:47
entries are the same and then the the
4:49
next sub diagonal one three six ten
4:52
fifteen that's the triangular numbers
4:53
which are again appearing down the um
4:56
down that
4:57
actually is
4:58
labeled second second column and that's
5:01
a a key feature here so another way
5:05
another way of generating
5:09
sequences
5:10
from this
5:12
is
5:13
to consider
5:14
sub diagonals
5:17
and in this case
5:20
this gives us
5:23
nothing
5:24
essentially
5:26
new
5:28
except for
5:32
except for a
5:35
a different kind of staggering
5:39
and that's worth exploring so see what
5:42
we would get if we just looked at the
5:44
sub diagonals let me
5:46
put it over here maybe i'll make some
5:49
spaces in here
5:51
okay so over here
5:52
uh this this first sub diagonal starting
5:54
with one two three we could get at that
5:56
by by taking this thing and removing it
6:00
i'm gonna copy it there
6:02
and maybe add a ten
6:05
okay i could uh
6:07
get the next one
6:08
copying that
6:10
i have to go down two i suppose
6:12
and then what's the next one we add
6:13
seven we add eight we add nine so it'll
6:15
be
6:16
45
6:18
and then we add 10 so it'll be 55.
6:21
okay and then we could keep going so um
6:23
for this it might be
6:24
convenient to have a bigger version of
6:26
this and with scientific workplace i can
6:28
create that pretty nicely by just to
6:31
creating a function so if i define f of
6:33
i j equals i minus 1 j minus 1 is
6:37
binomial coefficient so i'm
6:39
adjusting for the fact that scientific
6:41
workplace wants to start its indexing
6:43
with one okay and we want the pascal
6:47
array to start indexing with zero so
6:49
that's why there's a shift there so if
6:51
we define this function here and then i
6:54
can just use the
6:56
scientific workplace
6:58
capability matrix fill matrix
7:03
how about 18 by 18. 18 by 18
7:07
defined by function
7:08
f
7:09
okay so there's a bigger version of it
7:12
so for example if i want the next
7:14
column here the ones starting with 1 4
7:16
10 10 i could go down here
7:19
and take i think 10 of them 2 4 6
7:23
8 10.
7:32
yes
7:33
all right let's just do one more i'll
7:35
grab the first 10 of the next one
7:37
2
7:38
4
7:39
6 8 10.
7:42
okay
7:48
okay and then there's uh more stuff
7:50
which i will
7:53
just express by leaving it blank okay
7:56
so this this matrix here that i'm
7:59
creating is another variant
8:02
of the pascal
8:04
matrix you can see that now the usual
8:06
rows are appearing as cross diagonals
8:08
one one one two one one three three one
8:10
one four six one and this is certainly
8:12
also a useful kind of thing to consider
8:14
but from this linear algebra point of
8:16
view it's actually quite different
8:18
because this is a lower triangular
8:20
matrix and this one is
8:23
well
8:24
a regular one okay it's not it's not a
8:27
lower triangular that will mean that
8:29
defining the inverse of this on maxwell
8:33
will be considerably more problematic
8:35
than defining the inverse over here here
8:37
the inverses is
8:39
quite well defined as we've observed
8:41
before here it's it's much more
8:43
problematic so even though it may look
8:45
like the same information from a linear
8:47
algebra point of view from a matrix or
8:49
maxwell theory point of view these are
8:50
actually quite different
8:52
configurations
8:57
now once we have this
8:59
this idea in mind we can actually think
9:02
in
9:03
in another kind of bigger way
9:05
and
9:07
so
9:08
you see we have this
9:10
big two-dimensional array and what we're
9:13
doing when we
9:14
we're looking at the previous examples
9:16
is we were finding sequences
9:19
inside here by going across the rows or
9:21
going down the diagonals or going down
9:23
the sub-diagonals
9:24
we could also do other things we could
9:26
also for example go
9:28
down across diagonals
9:30
okay so let's have a look at what
9:32
happens if we do that
9:35
okay so i'm going to look at cross
9:37
diagonals and i'm going to start with
9:40
the diagonal entries for each one of
9:42
these diagonal entries i'm going to go
9:44
down to the left and pick up the little
9:46
sequence so i start with just one and
9:49
then the next one is one one
9:51
and the next one is one three one
9:54
and the next one is one six five one
9:58
okay
9:59
the next one will be one ten fifteen
10:02
seven one and um i can sort of arrange
10:05
those sequences those cross diagonal
10:08
sequences in this matrix here okay so
10:11
i'm putting them down now as rows
10:14
trying to organize them so that i can
10:16
see them 1 one one one three one one six
10:19
five one one ten fifteen seven one one
10:23
fifteen thirty five twenty eight nine
10:25
one
10:26
when we do it this way we are insured of
10:28
getting another
10:29
triangular matrix with pleasantly ones
10:32
along the diagonal
10:34
now
10:35
the online encyclopedia is
10:37
is quite aware of the importance of
10:41
two-dimensional arrays okay and
10:43
extracting sequences from them
10:46
and an important uh sort of technique
10:49
that that's used there is to take a
10:52
triangular matrix like this and encode
10:54
it as
10:55
a sequence
10:56
okay so how can you do that well just
10:58
sort of by reading this in the obvious
11:00
way you just sort of read it and ignore
11:02
the zeros you read one
11:04
one one
11:05
one three one one six five one et cetera
11:09
okay and if you know that it's going to
11:10
be triangular then you can recover the
11:13
triangular maxill from this sequence so
11:16
this sequence encodes
11:18
actually the entire information of this
11:20
maxwell although it's not displayed in
11:22
perhaps it's
11:24
constructive a fashion
11:26
so
11:27
is this sequence appearing well yes it
11:30
is not surprisingly um but it's
11:32
interesting that um
11:34
we can already see that
11:36
coefficient arrays for example so one
11:38
one of the
11:40
places where this happens
11:42
is to consider the coefficient array for
11:44
so-called morgan voice polynomials so
11:47
there are these morgan voice polynomials
11:50
which
11:52
if we write them out they're going to be
11:54
given by these coefficients along these
11:56
rows
11:57
so this one here might be like 1 plus 3x
12:00
plus x squared the next one 1 plus 6x
12:02
plus 5x squared plus x cubed okay so
12:07
those are
12:09
some interesting polynomials i don't
12:10
actually know
12:11
the context in which they arise but i
12:13
think it'd be quite interesting to find
12:14
out so what what we're seeing is that
12:16
we're we're touching base with you know
12:19
yet another aspect of combinatorics or
12:21
algebra
12:23
in this fashion but there are some other
12:25
um interesting things here there's also
12:28
an interpretation in terms of dyke paths
12:31
if you
12:32
become a fan of the online encyclopedia
12:34
and they urge you to all become fans
12:36
then the story of dyke paths
12:40
naturally ends up playing quite a bit a
12:41
big role and so you end up being quite
12:43
motivated to figure out you know what
12:45
the story is with dyke paths because
12:47
that's
12:48
that makes a connection with all kinds
12:49
of sequences
12:52
all right so this is pretty interesting
12:54
but it's not really the only thing that
12:56
we could do right because
13:01
because if we go back to our
13:04
pascal
13:05
matrix what we were doing here is we
13:07
were looking at these cross
13:08
diagonals starting from
13:10
the
13:11
diagonal position but there are other
13:13
cross diagonals which don't start there
13:16
which start at the next sub-diagonal the
13:19
one starting with one two three four so
13:21
that would be the
13:23
the sequence with first one entry one
13:25
and then two one and then three four one
13:28
and then four ten six one and then five
13:32
twenty twenty 8 1 and so on
13:35
so
13:36
because we're trying to systematically
13:38
look at what's going on here right we're
13:40
we're not prejudiced as to you know
13:42
what's important and what what's not we
13:44
want to put everything on the table in
13:46
some kind of fashion okay and sort of
13:48
organize
13:50
things organize our understanding
13:51
organize the structure of things so
13:53
let's record that all right so
13:57
that's what
13:58
happens here so another possibility look
14:00
at the other cross diagonal so here's
14:01
the we've repeated the matrix so you can
14:03
see it a little bit and then
14:05
over here
14:06
first cross diagonal one and then the
14:08
next one is two one and going down here
14:12
uh
14:13
next one three four one and then four
14:14
ten six one et cetera we got another
14:17
triangular matrix its corresponding
14:19
sequences this one one two
14:21
one three four one
14:23
four ten six one etcetera get this thing
14:25
here and then we look that up and
14:27
happily it's also there
14:29
and
14:32
what do they say about it um coefficient
14:34
array for morgan voice polynomials okay
14:36
so it's still connected with those
14:38
things here and there's a number of
14:40
compositions now
14:43
preserving full transformations of an n
14:45
chain of height k
14:48
okay this is a
14:55
sub triangle there's one here that's
14:57
quite interesting
14:58
right down here in the bottom i don't
14:59
know if you can read it but i'll read it
15:01
it says from our bagula's comment in
15:04
a053122
15:07
cf damn you know link page 10. this
15:10
array gives the coefficients mod sine
15:13
of the characteristic polynomials for
15:15
the cartan matrix of the root system a n
15:19
so the carten matrix of the root system
15:21
a n well that's something i'm kind of
15:23
interested in because that's very
15:25
closely connected to the other series
15:28
that i'm
15:29
creating on
15:31
the dynamics on graphs
15:33
and the mutation game the numbers game
15:35
etc and carton matrices figure very
15:38
prominently here so we're talking about
15:39
characteristic polynomials for a carton
15:42
matrices so this is another way of
15:44
generating a list of interesting
15:46
polynomials because the root systems a n
15:49
are indexed by a natural number n
15:52
okay so uh that's great um until we get
15:55
this all these coincidences are always
15:57
appearing here right you you just feel
15:59
as if you're touching base with more and
16:01
more things and there's all these things
16:02
to to learn about and and and to
16:04
understand and and and more and more
16:07
connections that are waiting to be
16:09
uncovered
16:11
this way this is such a great area for
16:13
amateurs right you can become expert in
16:16
any one of these things and you can
16:18
become a leading investigator in in some
16:21
aspect of this this remarkable story
16:27
okay
16:28
so
16:30
what about inverses all right
16:32
so
16:33
one of the themes that i'm trying to
16:35
convince you of is that it's not just a
16:36
question of having these arrays
16:39
that we want to when possible think
16:41
about the arrays as actually having
16:43
linear algebra content
16:45
in other words that they're
16:47
maxels
16:48
or
16:49
matrices
16:51
in particular that we can multiply them
16:53
and that we have the potential of taking
16:56
inverses
16:57
and i'm hoping this is now almost a
16:59
second nature kind of thing for you to
17:02
think about oh there's an interesting
17:04
matrix i wonder what its inverse is
17:07
okay so here is the first of those
17:11
so cross diagonal matrices that we
17:13
cooked up okay and let's calculate its
17:15
inverse
17:18
here's the inverse
17:19
it's also triangular it's got ones in
17:21
the diagonal that's not a surprise uh
17:23
the entries in the
17:25
sub diagonal are minus are just the
17:26
negatives of these ones that that
17:28
typically always happens i'm just not
17:29
surprised and then the other ones are
17:31
different okay they're all going to be
17:33
natural numbers that's not a surprise
17:35
um oh actually not they're all going to
17:38
be integers okay that's not a surprise
17:40
but when we actually look at the signs
17:42
here then i think this is quite curious
17:45
because
17:46
these
17:48
the signs here are all appearing on what
17:51
we might call the odd positions if we
17:53
think of this as a
17:54
chess board or something with say this
17:56
top corner black okay
17:59
so so that's the one one entry that's
18:01
the two to entry so the the black ones
18:04
are all the let's say even entries um
18:06
the ones where the sum of the
18:07
coefficients is is even
18:11
that definition is not going to change
18:13
if you change your indexing from
18:15
starting with zero to starting with one
18:18
so
18:19
what's happening here is that
18:21
we're getting signs at all the odd
18:24
places though
18:26
the positions that would correspond to
18:28
red squares in this checkerboard right
18:30
right all of those and then all this
18:33
third sub diagonal and then the fifth
18:35
sub diagonal the seventh sub diagonal
18:38
down here
18:40
now
18:41
when i saw this i was surprised
18:43
why should this be
18:45
you've got this matrix okay it's it's a
18:47
pleasant matrix it comes from the
18:50
pascal
18:51
matrix quite pleasantly but why should
18:53
it have this seemingly quite a strong
18:56
property
18:57
and so i i asked myself well this this
19:00
certainly can't be automatic right i
19:02
just just checked on that so let's just
19:04
let's just check that ourselves again so
19:06
i'm going to take that same matrix and
19:08
i'm just going to modify one of them
19:10
let's modify that
19:12
that five okay let's turn that five into
19:14
a six all right let's do that
19:19
and now let's observe the signs
19:26
the signs still appear
19:28
to be in all of the odd places
19:32
yes indeed
19:34
okay what if i change that that 6 into a
19:36
26
19:38
get rid of this one
19:46
we're still getting
19:52
um
19:53
i'm still getting the signs at all the
19:55
odd places am i right
19:58
yes it would appear so
20:03
okay that's kind of cool what happens if
20:05
i change
20:06
it change this to
20:08
99
20:17
ah now something's
20:20
difference happened
20:21
right now we've broken this pattern of
20:24
we still have signs down this first sub
20:26
diagonal the second sub diagonal is
20:28
positive the third sub diagonal is not
20:30
quite negative anymore it's negative up
20:32
to a point and then there's positive
20:33
entries etcetera here we have two
20:38
here we have a negative sign plus a sign
20:40
plus plus plus so it's no longer having
20:43
this checkerboard sign aspect and so i
20:47
had to i had to make a couple of changes
20:49
to sort of see that happening right the
20:51
first sort of change that i made
20:52
actually the first two changes that i
20:53
made seemingly didn't have that such an
20:55
effect
20:56
that checkerboard pattern in the science
20:59
was still there
21:01
okay um
21:02
look this seems pretty interesting to me
21:04
um one should try to
21:06
understand
21:09
what
21:10
what is responsible for this strong
21:12
property this seemingly strong property
21:14
of having a matrix which starts off just
21:17
with natural number entries and you're
21:19
taking its inverse and you're getting
21:21
this checkerboard pattern of signs
21:23
okay
21:24
and
21:26
okay and you know
21:28
why was this unchanged when i made um
21:31
one
21:32
change to the entry but then when i made
21:35
the second change
21:36
um involving a second square then we saw
21:39
the pattern disrupted
21:41
i don't know
21:42
and what about what about this um this
21:45
inverse that we're getting right i mean
21:47
you know
21:48
this inverse if we ignore the signs uh
21:50
it has
21:52
sequence 1 1 1
21:54
2 3 1 5 9 5 1 etc we can look that up
21:59
do we have anything there
22:03
triangle formed from even-numbered
22:06
columns of triangle of expansion of
22:08
powers of x to terms of chebyshev
22:11
polynomials u n of x is the second kind
22:15
cool
22:15
so now we're finding out that
22:18
this triangle it's actually you know the
22:20
triangle the online encyclopedia is
22:22
recognizing this is not just a sequence
22:24
this is really coming from a
22:25
two-dimensional story that this is this
22:28
two-dimensional story is something like
22:30
a change of basis matrix from
22:33
um
22:34
powers of x to chebyshev polynomials of
22:37
the second kind but that's interesting
22:38
for us because we're interested in the
22:40
chebyshev polynomials um so that that's
22:42
uh quite cool and it's kind of neat that
22:45
you know that
22:46
the inverse the inverse actually has
22:49
some content as well it's not we're not
22:51
just going for something that's really
22:53
structured to something that was random
22:55
if we were getting that then we would
22:57
say well this matrix point of view is
22:58
not really that helpful but here we're
23:01
not getting something random that's
23:02
telling us that this matrix point of
23:04
view is still quite apt it's still quite
23:07
something interesting
23:09
okay what happens to the other one
23:11
remember there was a second one that we
23:12
looked at this one here when we take its
23:15
inverse
23:17
we get the same checkerboard
23:20
pattern of signs which again is
23:23
surprising to me you know why should
23:24
this be
23:25
and their corresponding sequence when we
23:28
write that down we can also find that
23:30
and not too surprisingly that's also
23:33
um connected with uh this change of
23:37
basis um
23:39
powers of x to the chebychev
23:42
polymorphials and there's other things
23:43
here that one should you know look at
23:45
and um
23:47
and contemplate
23:49
so i'm hoping that i've convinced you
23:51
that there's all kinds of interesting
23:52
connections between this two-dimensional
23:55
maxwell sort of approach that we're
23:56
taking and the online encyclopedia and
24:00
you know the online encyclopedia
24:02
recognizes that
24:03
but
24:04
if you take
24:05
one of these just one of these
24:08
maxwells say say that one there right
24:11
it conceivably can generate all kinds of
24:14
sequences it's not at all clear that
24:16
every single column or row is going to
24:18
well obviously not everyone but you know
24:20
that even the first five or or so
24:23
rows and columns of this thing here are
24:25
going to be in the online encyclopedia
24:27
and then if we start making uh further
24:30
you know progressions uh you know going
24:32
down
24:33
uh sub diagonals or across diagonals uh
24:36
then we get other other things so
24:40
it very well could be that
24:43
what we should be thinking about is some
24:45
kind of augmentation or maybe a
24:48
actually a separate project where
24:51
a team of people
24:52
start thinking about two-dimensional
24:56
integer
24:57
arrays or maxels
25:00
and
25:02
and
25:04
sort of investigating you know
25:06
what do these things look like
25:09
what are the interesting ones and and
25:11
having a sort of a mapping so that we
25:13
can quickly identify
25:15
whether a given sequence is is occurring
25:18
in one of these maxwells so this really
25:21
requires some some ai right but you
25:23
can't do this by hand or optimal or
25:25
totally uncertain
25:27
you really need some ai to to clarify
25:31
the the
25:32
the relationships between all the
25:34
different sequences that you could get
25:36
by you know by walking down uh
25:39
these um these maxels in different ways
25:42
so i think there's uh
25:44
some interesting discussion to be had
25:46
here about about potentialities that
25:48
might interest computer science or
25:51
programming types uh i think
25:54
this this is really a powerful
25:57
thought direction