Basic Organization of 6 objects
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We continue our introduction to this elementary course in Pure Mathematics, concentrated on the magical and beautiful properties of the number 6. In our first video we introduced Nodes, Edges and Meets. In this video we review these and give a simple geometrical interpretation of them in terms of planar geometry, just involving points and straight lines. We also introduce Triangles and more complicated objects called Pairs, which are just 2-sets of Triangles satisfying a certain property. Already the counting aspects are getting interesting! Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated.
00:00
[Music]
00:09
hello
00:10
and welcome to six i'm norman bober and
00:13
this is an elementary
00:15
course in pure mathematics meant for a
00:18
very broad
00:19
audience in this our second lecture
00:22
in the course we are going to continue
00:26
with the basic organization of
00:29
fundamental objects associated to
00:32
six things so let me remind you about
00:35
the
00:36
conventions that we've established so
00:37
far we've introduced first of all the
00:39
nodes
00:41
and there are six of them and they are
00:43
the fundamental objects
00:44
in the course and these are the numbers
00:47
one two three four
00:48
five six arranged in this familiar order
00:52
from the nodes we first then created
00:55
edges
00:56
what is an edge an edge is a two set
00:59
of nodes that means it's a set
01:03
consisting of exactly two nodes
01:06
and this is using wild burger convention
01:09
for
01:09
data structures where we do not use
01:12
commas
01:12
unless we intend to organize the
01:14
elements of a data
01:16
structure so just having spaces alerts
01:19
us to the fact that this is an unordered
01:21
object we can move these things around
01:22
that's a set
01:24
now instead of having to write all of
01:26
this out every time we want to talk
01:28
about a two set
01:29
we shorten that by agreeing that we're
01:32
going to represent this two set with
01:34
just
01:34
the expression one followed by two
01:38
in other words we take the elements in
01:39
the set and we put them together
01:42
but with the agreement that the smaller
01:44
one in the ordering
01:46
comes first that way we get a canonical
01:49
representative
01:50
for every edge and then here
01:54
are the 15 edges from one two
01:57
all the way to five six
02:00
and then we introduced the idea of a
02:02
meat what was a meat
02:04
a meat was a two set of non-incident
02:08
edges first of all we say that two edges
02:12
are
02:12
incident say like 2 3 and 2 4 when they
02:15
share a node while 1
02:18
6 and 2 4 are non-incident because they
02:21
don't
02:22
share a load they don't overlap they
02:23
don't intersect as sets
02:26
so what a meat is is a two set of
02:28
non-incident edges
02:29
for example this thing here this is a
02:31
meat
02:32
it's a set there's no commas it's just a
02:36
space between this object and this
02:38
object
02:39
this object is itself an edge
02:42
this is another edge the short forms for
02:45
these things we've agreed
02:47
are just one two and three four so
02:49
another way of representing this
02:50
meat in more compact form would be this
02:53
it's a set consisting of the edge one
02:55
two and the edge three four and then we
02:59
agree to
02:59
further compress our notation to
03:02
write it as one two three four
03:05
and our convention here is that
03:09
we're going to take these two edges and
03:11
we're going to put the
03:13
the one that starts with the smaller
03:15
number
03:16
first okay so we know these two edges
03:20
by agreement are disjoint that means
03:22
they don't have any common nodes
03:24
so amongst the four there is a smallest
03:28
node in this ordering in this case one
03:31
so we put the edge with the one
03:33
to the left and the other edge to the
03:35
right
03:37
when we do that we get a canonical
03:39
representative a unique way
03:41
of writing a given meat
03:47
now notice that in this particular
03:48
example the two ingredient
03:50
edges one two and three four are both
03:54
not incident with a unique
03:57
other edge there's another edge sort of
04:00
off
04:01
the page which is the edge five six
04:05
these two edges are not incident with
04:08
five six and five six is the only edge
04:10
that has that property
04:14
so in my first lecture i gave you a
04:16
homework exercise
04:17
which was to list all 45 meets
04:21
therefore verifying that there are
04:23
indeed 45 of them as i've claimed
04:27
now i'm going to present a solution to
04:29
that homework
04:30
which maybe is the one that you've
04:33
obtained but maybe also different
04:35
i'm going to use that observation that
04:37
we just made
04:38
that given any meat
04:42
there's a unique edge which does not
04:46
overlap or is not incident with either
04:48
of the
04:49
edges contained in that given meat so
04:52
that means we can kind of
04:53
organize all of the meats by the
04:56
associated edges
04:58
which are not contained in them
05:02
so we're going to do that by then
05:04
listing the edges
05:05
and i'm going to reverse the order of
05:07
the edges starting with 5
05:08
6 because that means that i end up
05:11
listing the actual
05:12
meats in a maybe a more natural way so
05:16
what am i doing here
05:17
so associated to the edge 5 6 i have
05:20
three different meats
05:24
what are they they are simply the meats
05:27
involving the
05:28
four nodes which do not appear
05:31
in the edge five six in other words the
05:34
nodes one two three four
05:36
now how many meats can we create from
05:39
those four
05:40
nodes well there's exactly three of them
05:43
these three here
05:45
this is a very important point if we
05:48
have four objects and we're interested
05:50
in
05:51
grouping them into two groups of two
05:53
each
05:54
like spark okay there's a group of two
05:57
there's a group of two
05:58
that's one way of subdividing four
06:01
objects into two groups
06:02
or two sets of two elements each
06:06
and there's exactly three ways of doing
06:08
that
06:10
three ways of grouping four objects into
06:13
a pair
06:14
of pairs we see that right here
06:17
the four meats are one two three four
06:21
one three two four and one four two
06:24
three
06:28
once we've understood that then it
06:30
becomes
06:32
almost completely clear how to organize
06:34
these things it becomes also clear
06:36
why there are 45 of them because we know
06:39
there are 15
06:41
edges and what we're doing is we're
06:43
associating
06:44
three meats to every edge so we're going
06:47
to get 3 times 15 which is 45 metes
06:51
so for example associated to the edge 4
06:54
6
06:55
the other nodes not involved in that are
06:58
1
06:58
2 3 and 5. and here are the
07:02
three meats that we can create from
07:04
those four nodes
07:07
one two three five one three two five
07:11
and one five two three
07:14
you'd say well wait a minute what about
07:16
you know two three
07:18
one five like changing those things
07:21
around
07:21
well we're not allowed to do that
07:23
because we've agreed that when we
07:24
represent this thing
07:26
the smallest note has to appear on
07:29
the extreme left okay so there's only a
07:32
unique way of writing each one of these
07:34
things
07:36
okay and then here is the list now you
07:38
can check
07:39
hopefully that i've got them all right
07:40
so for example for 1 4
07:42
the nodes not involved in one and four
07:44
are two three five and six and here are
07:46
the
07:47
three meats created from them two three
07:50
five six
07:52
two five three six and two six
07:55
three five 15
07:59
meats created from our six
08:02
fundamental objects now i want to
08:05
emphasize that the fundamental objects
08:06
that we're dealing with and that we're
08:08
constructing
08:09
as we go are purely combinatorial
08:13
in essence there's really no arithmetic
08:15
involved in fact
08:16
no higher mathematics beyond really
08:19
primary school
08:22
familiarity with numbers and this basic
08:24
idea of grouping things
08:26
together in convenient ways
08:29
nevertheless as human beings we
08:32
often like to understand things
08:36
using a visual representation we are
08:39
often quite comfortable once we have a
08:42
visual representation of some abstract
08:44
thing
08:45
because our brains have been
08:47
biologically you know oriented
08:49
to processing visual information
08:52
very well so we're very good at that
08:56
and so it makes sense for us actually to
08:58
try to find
08:59
visual representations of mathematical
09:03
situations and we can do that here and
09:06
when we do that
09:07
you'll see perhaps why i've chosen some
09:09
of the notation
09:11
that i've chosen okay because it's
09:13
aligned
09:14
with this geometrical manifestation of
09:18
the theory i'm going to show you now
09:20
okay so let's work in the plane which is
09:23
very convenient for visual
09:25
representation because we can draw it on
09:27
a piece of paper
09:28
and the three-dimensional complexity is
09:30
not there
09:31
so we're just working in the plane and
09:34
we start with our six
09:37
nodes which we represent as points
09:41
on the page one two
09:44
three four five six and i've scattered
09:48
them roughly around a kind of a circular
09:50
region
09:51
but that's rather unnecessary we could
09:53
have put them anywhere
09:55
i just like it this way because it's
09:57
sort of roughly symmetrical
09:58
but it shouldn't be thought of as a
10:00
regular hexagon it's just
10:02
six points in the plane and there
10:06
are the nodes then what are the edges
10:10
well the edges are given by pairs of
10:13
nodes two set of nodes for example the
10:16
two set 2 3 or the
10:20
edge 2 3. that can be then represented
10:24
by the line that is determined by
10:28
the nodes 2 and 3. namely this one right
10:30
here so this line that i've drawn
10:33
is a visual representation of what we
10:35
called an edge
10:36
it's actually the edge maybe i'll put it
10:38
right here two
10:40
three okay that's the name for
10:44
this edge
10:48
now this is only one possible way of
10:50
representing it maybe somebody else
10:51
might say oh i don't want an entire line
10:53
i'll just do a line segment from three
10:55
to two that would be another
10:57
also valid of
11:00
the edge two three not quite as useful
11:02
as
11:03
uh this though okay
11:06
and uh you know so we could do that with
11:08
lots and there's there's two
11:10
six okay um
11:13
you know there is uh one three maybe
11:16
i'll put up here one
11:17
three okay so we can see there's lots of
11:20
um
11:21
edges and can we count that there's
11:22
actually 15 of them
11:24
well let's see um so
11:28
through say this one here we have one
11:31
two
11:31
three four five we have five
11:35
um lines passing through that one
11:39
and count there's also five passing
11:41
through that one five there
11:42
five there and five and five there so
11:44
all together if we count
11:45
via the nodes we get six times five or
11:49
thirty but thinking about it for a
11:52
second we see that that counts
11:54
a line twice for example this line here
11:56
1 3
11:57
will be counted once at the node 1 and
12:00
also another time at the node 3. so we
12:03
have to take that 30 and divide by
12:06
2 to get 15. so that sort of
12:09
is a verification that we've got 15
12:10
lives without actually having to to go
12:12
through them although we could easily go
12:14
through them also
12:16
okay what about meats well as you may
12:19
have guessed
12:20
there are some additional information on
12:22
the diagram namely these little
12:24
black dots which are
12:27
mates of edges so for example this one
12:30
right here
12:31
okay is the meat of
12:34
this edge 2 6 and this
12:38
edge 1 3 it's actually a physical meat
12:40
like an intersection
12:42
in geometry so this is a representation
12:46
of what we called a meat which meat is
12:49
it
12:50
well we have line one three or the edge
12:53
one three
12:54
and the edge two six so the meat here is
12:57
one three dot
13:01
two six
13:06
and we could easily give uh names to the
13:09
other ones for example let's do this one
13:10
here
13:11
okay it's passing through four six and
13:14
three five so its name would be three
13:16
five
13:17
dot four 6.
13:23
now we said there should be 45 of those
13:25
can we see 45 little black dots
13:28
no we can't and why not well because
13:31
some of them are going to be off the
13:33
page so for example
13:35
you know if you take this line here and
13:38
this line here
13:39
now there's the line 4 5 and the line 3
13:41
2 or i should say the edge
13:43
4 5 and the edge 2 3 then their meat is
13:47
going to be up there somewhere so we
13:48
would have to extend the paper
13:50
to be able to see the black dot that
13:52
should be drawn there
13:56
now if some of the lines are almost
13:58
parallel like these two here are
14:00
almost parallel maybe but maybe not
14:02
quite we might actually have to go quite
14:04
far out in order to find
14:07
the physical meat of those two
14:10
edges or lines in fact we could see that
14:14
if we had cooked things up so that you
14:17
know two of these lines were
14:18
exactly parallel then we might have a
14:20
little bit of a problem
14:22
because then the the meat that we would
14:24
like to draw would be actually very very
14:26
far away in fact it wouldn't exactly be
14:28
on the
14:29
on the page even if we extended it
14:31
arbitrarily
14:34
suggesting that you know in order to
14:36
really make this kind of diagram we may
14:38
have to go to something called
14:39
projective geometry
14:41
which allows us to augment our
14:43
understanding with points at infinity
14:46
and we'll be talking about that a little
14:47
bit further on as we delve further into
14:49
the course
14:51
but nevertheless at this stage here we
14:53
see that we have a nice representation
14:56
of nodes and edges
15:00
and beats as geometrical objects in the
15:03
plane
15:04
so we have this purely combinatorial
15:07
description in terms of sets of things
15:10
and here's a corresponding geometrical
15:14
visualization of that combinatorics
15:18
so while we have this picture here
15:20
perhaps it's a time to ask well what
15:22
would be the sort of the next
15:23
natural step for us to investigate
15:26
maybe the geometry will naturally lead
15:30
us
15:30
to asking some decent questions as to
15:33
what to study next how to enlarge our
15:36
landscape
15:38
so if we look at the story we've talked
15:41
about
15:41
pairs of points two sets
15:44
and two sets of two sets
15:48
what about taking three nodes
15:51
that's a kind of natural extension of
15:54
looking at
15:55
pairs of notes so for example we could
15:57
look at
15:58
the set 2 3 4
16:02
that set of points and how could we
16:06
represent that set with a geometrical
16:10
object
16:11
well one thing that is reasonably
16:13
natural to do is to consider
16:15
the triangle formed by those three
16:19
nodes this triangle right here
16:28
so that's a geometrical representation
16:31
of the set consisting of two
16:36
four and three
16:42
as i said it's unordered so i don't have
16:43
to worry about the order of the elements
16:48
there
16:50
okay so that's a natural object
16:53
and once we've drawn that one if we look
16:56
back and say well actually once we've
16:59
singled out
17:00
three nodes to form a triangle like this
17:04
then there's also kind of the opposite
17:07
or the remaining
17:08
nodes in this case 1 5 and 6.
17:12
so they are also going to form a
17:15
triangle
17:16
a triangle which is in in some sense um
17:20
disjoint from from this one it doesn't
17:22
share
17:23
any notes let me just draw that also
17:27
here's the triangle five
17:30
six one okay
17:35
that's the set five
17:38
six one
17:41
so when we're looking at triangles it's
17:44
not
17:45
unreasonable to not just consider just
17:48
single triangles but also these
17:50
pairs of triangles that seem to want to
17:53
fit together so that's actually a very
17:56
natural and profitable
17:57
kind of direction to to think about all
18:00
right so let's introduce
18:02
triangles so a triangle is a
18:05
three set of nodes and there's i claim
18:09
20 of them our notation is that well
18:13
here's a three set
18:14
consisting of the nodes one two and
18:16
three and we're going to
18:18
adopt a very similar notation as we did
18:20
for edges
18:21
so instead of having to write these
18:23
braces and
18:24
these objects with spaces we'll just put
18:26
them together
18:28
as one two three with the agreement
18:31
that we're going to choose the ordering
18:34
imposed by the natural order of our six
18:37
objects so the smallest one comes first
18:40
and then the next smallest one and so on
18:44
that gives a canonical representation
18:47
of our our triangle
18:51
and how many triangles are there well
18:55
in combinatorics this is a
18:58
number called a combination okay we're
19:01
looking at
19:01
how many ways we can choose
19:05
uh three things from six
19:08
so six choose three it's a binomial
19:12
coefficient
19:13
and explicitly as an expression
19:15
involving natural numbers
19:17
it's six times five times four divided
19:20
by three
19:21
times two times one and this denominator
19:24
6 cancels with that 6 and we do indeed
19:26
get 20.
19:28
now let me say a little bit about this
19:30
formula in case it's maybe not familiar
19:32
or you've forgotten it
19:33
okay so we want to count triangles like
19:36
this the
19:38
numerator and the denominator both have
19:40
a meaning
19:41
okay the numerator is counting the
19:44
number of
19:46
three element lists or three lists that
19:49
we can create
19:50
from six things okay so
19:54
if we're creating a list like something
19:55
like this where there's actually a first
19:57
object and a second object and a third
20:00
object out of a total of six
20:03
then there's six choices for the first
20:06
one and once we've
20:08
chosen that one there's five objects
20:10
remaining to be put
20:12
in the second place and after we've
20:14
chosen that one there's
20:15
four objects remaining to be put in the
20:18
third place
20:19
so that's all together six times five
20:21
times four possible
20:22
three lists from six
20:26
objects are nulls
20:30
now why do we divide by 3 times 2 times
20:34
1
20:35
because that represents the number of
20:37
ways of rearranging
20:39
three things like once we have chosen
20:42
three objects out of the six
20:44
say objects one two and three then
20:46
there's six
20:47
possible ways of permuting them or
20:50
rearranging them
20:51
in other words when we write down all
20:54
the lists
20:56
created from one two three there's
20:57
actually one two three
20:59
one three two two one three two three
21:02
one
21:03
three one two and three two one there's
21:05
six different lists
21:06
that we could create from just those
21:08
three nodes
21:11
and the number is six because well we've
21:13
got three things and we can put the
21:15
first one we have three possibilities
21:17
for that and then there's two
21:19
possibilities for the second one and
21:20
then there's one possibility for the
21:22
third
21:25
so this is why we have to take the
21:27
number of three lists and we have to
21:28
divide by the number of ways of
21:30
rearranging
21:31
any one of those uh lists to make an
21:33
equivalent
21:34
set that's why this is the expression
21:38
and why we do get 20. and we can verify
21:40
that it's correct by
21:42
listing them all one two three one two
21:45
four one two five one two six
21:49
one three four one three five one three
21:51
six one
21:52
four five one four six one five six
21:55
two three four two three five two three
21:58
six
21:58
two four five two four six two five six
22:02
three four five three four six three
22:04
five six and four five six
22:07
and make sure that you agree that this
22:08
is a very natural way of listing
22:11
these 20 objects in a way it's like the
22:14
mathematical equivalent
22:16
of alphabetical order lexigraphical
22:19
ordering in mathematics
22:21
where we sort of like count but
22:24
we we increase the right most thing like
22:27
a speedometer
22:28
also we're sort of like we're counting
22:30
and then once we've gotten to the max of
22:32
that one
22:33
then we we go back and we we augment
22:36
the next one to the left
22:40
so uh there are 20 triangles
22:44
formed from our six notes a bit of
22:47
additional notation
22:48
we'll say that say a triangle say one
22:50
two three is
22:51
incident with the nodes one two and
22:54
three
22:55
and that it's also incident with the
22:58
edges
22:59
one two one three and two three
23:02
so these are the nodes which are
23:04
contained in this triangle because after
23:06
all it's actually a set
23:08
okay and these are the two sets
23:12
which are subsets of
23:15
the set corresponding this triangle
23:18
triangle one two three is really the set
23:20
consisting of the objects one two and
23:21
three and this
23:23
is really the set consisting of the
23:25
objects one two so that's a
23:26
subset of this set
23:30
so there's a natural association between
23:33
a triangle and it's three nodes and it's
23:36
three
23:37
edges now if we have two different
23:41
triangles
23:42
triangle here triangle there there's
23:44
various relations that they can have
23:46
okay one relation we can say that
23:49
they're incident
23:50
precisely when they share exactly one
23:53
node
23:54
so geometrically it's sort of like this
23:55
there's one triangle
23:57
there's another triangle but they share
23:59
a node
24:00
like this okay
24:04
and we say that a pair
24:08
this is a different notation now a pair
24:11
is a two set
24:13
of non-intersecting triangles in other
24:16
words triangles which don't overlap at
24:18
all which
24:19
share not just one node or even two
24:22
nodes they're completely disjoint
24:24
like the two triangles i showed you in
24:26
that diagram
24:29
so for example here
24:33
the triangle one two three
24:37
and together with the triangle four five
24:39
six
24:40
so here are two triangles this is then a
24:42
two set
24:43
of triangles and these are indeed
24:45
non-intersecting because they don't
24:47
overlap at all there are no common nodes
24:52
and now this is going to be quite an
24:54
important object for us a pair
24:56
so let's agree to have a short
25:00
notation for it we're going to do the
25:03
same kind of thing that we did with
25:04
meats
25:05
we're going to arrange the two triangles
25:09
together with a dots separating them
25:13
but the triangle that's going to come
25:15
first is the triangle that has
25:17
the smallest node
25:22
now since we have two triangles actually
25:24
all six nodes are being used here right
25:26
because these are disjoints so
25:28
all six numbers are appearing here so
25:30
the number one has to appear
25:31
somewhere so in fact this leftmost entry
25:35
will necessarily be a one okay
25:38
so the left triangle is the triangle
25:41
that contains
25:42
one and its entries are listed in
25:44
ascending order
25:45
and then there's a dot and then the
25:47
other triangle
25:49
whose nodes are also listed in ascending
25:52
order so that the four is the
25:54
smallest one in that so this is a
25:56
canonical way
25:58
of representing a pair and a pair is a
26:01
two set
26:02
of non-intersecting triangles so now
26:05
we've introduced the idea of a pair
26:07
and i claim that there are in fact 10
26:10
pairs
26:14
that's maybe almost obvious because
26:16
we've seen that there are 20 triangles
26:20
okay and each triangle has a unique
26:24
other triangle which it's disjoint from
26:29
okay so those two triangles then
26:32
together form a pair
26:33
so we're taking the 20 original
26:35
triangles and grouping them into groups
26:37
of two
26:38
and so there's obviously going to be 10
26:40
such groups
26:42
or 10 pairs
26:45
and here they are one two three four
26:48
five six
26:49
one two four three five one
27:04
356-125-346-126-345-134-256-135-246-136-245-145236
27:05
three
27:06
one four six two three five
27:08
and one five six two three four
27:11
and i hope you'll see that this is sort
27:14
of an obvious thing that i've done i've
27:16
just
27:16
used this lexographical orientation to
27:20
organize things by sort of counting the
27:22
first
27:23
triangle successfully sort of from the
27:26
smallest one
27:27
all the way up to the biggest one
27:29
subject to the condition that it has to
27:31
contain a one
27:33
and then the rest is just the other
27:36
nodes arranged in
27:37
increasing order so here are the ten
27:42
pairs that are now part of our landscape
27:45
of six
27:48
now let's make another definition about
27:51
two triangles okay so if we had two
27:53
triangles we said that
27:54
they were incident when they shared
27:58
exactly a note i now want to also
28:01
introduce a second
28:03
term that captures when they
28:06
are sharing two nodes
28:09
like this
28:12
let's say that two triangles are
28:14
adjacent precisely when they
28:17
share exactly two nodes
28:20
let's think about triangles for a second
28:22
here's a triangle here's a second
28:23
triangle
28:24
there's various relationships possible
28:27
between these two triangles they could
28:29
be
28:29
incident they could be adjacent
28:33
or they could be in fact disjoint
28:36
there's three different
28:37
possible relations between two different
28:40
triangles
28:42
it's actually four relations if you
28:43
include the possibility that they could
28:45
actually be the same
28:49
now for your homework i want you to
28:51
investigate
28:53
the story with pairs the story with
28:56
pears is in some sense more attractive
28:58
than it is
28:59
with triangles because it turns out that
29:03
any two of these pairs
29:07
are related in exactly the same way as
29:09
any other pair
29:10
of pairs okay so what's the homework
29:14
show the following so here's a statement
29:16
given two pairs each triangle
29:18
in one is incident and adjacent with the
29:21
others triangles
29:22
and when i make a statement like this i
29:25
want you
29:26
to think about justifying the statement
29:28
or explaining
29:30
why the statement is true so i claim
29:32
this is a true fact
29:34
and your job is to write out write out
29:37
an explanation for the truth of this
29:40
fact
29:42
what in mathematics we call a proof okay
29:44
but it's just an explanation
29:46
so let me explain what's required here
29:48
it's actually
29:50
not that hard but it is a bit of a
29:52
mouthful so what we're saying is if we
29:54
take
29:55
two random triangles say this one here
29:57
and this one here
29:58
and we sort of look at how the triangles
30:01
inside those pairs are related
30:03
so this one here contains two triangles
30:06
this one here contains two triangles
30:08
so if we look at say one of the
30:10
triangles in this pair say 146
30:12
its relation with this triangle is that
30:14
it's
30:16
incident because it has exactly one note
30:18
in common
30:19
and with a relation with this triangle
30:21
is that it's adjacent it has exactly two
30:23
nodes in common
30:26
and what i'm claiming is that that
30:27
happens no matter which pair you take
30:29
and no matter which triangles you look
30:31
at
30:32
so the the collection of pairs is
30:36
somehow more symmetric than the
30:38
collection of triangles
30:41
okay there's a richer symmetry here with
30:43
all these bases
30:44
that the connection between any two
30:47
pairs is the same as a connection
30:48
between any other two pairs
30:50
so i want you to write this thing down
30:53
so we've had some more objects to
30:56
enrich in our landscape of six i'm
30:59
norman wahlberger
31:01
thanks for listening
31:06
[Music]
31:10
you