Views of the N-Cube and {MathILy, MathILy-Er} Math Jam

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AoPS Instructor and MathILy Director dr. sarah-marie belcastro leads students in exploration of the N-cube and answers questions about {MathILy, MathILy-Er}.

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Facilitator: sarah-marie belcastro

kguillet 2015-04-23 19:20:38
Welcome to the Views of the N-Cube and {MathILy, MathILy-ER} Math Jam! The Math Jam will begin at 7:30pm ET (4:30pm PT).
kguillet 2015-04-23 19:20:40
Please note that the classroom is moderated. This means that all your questions and comments go first to the moderators. We may or may not choose to share your comments with the whole room.
kguillet 2015-04-23 19:29:56
Hi everyone. We'll get started in just a couple of minutes.
kguillet 2015-04-23 19:30:55
Alright, let's get going!
kguillet 2015-04-23 19:31:04
Hello and welcome to the Views of the N-Cube and {MathILy, MathILy-Er} Math Jam!
kguillet 2015-04-23 19:31:08
Before I introduce our guests, let me explain about this classroom.
kguillet 2015-04-23 19:31:13
This room is moderated, which means that all your questions and comments come to the moderators. We may share your comments with the whole room if we so choose.
kguillet 2015-04-23 19:31:19
Also, you'll find that you can adjust the classroom windows in a variety of ways, and can adjust the font size by clicking the A icons atop the main window.
kguillet 2015-04-23 19:31:25
In this math jam, AoPS Instructor and MathILy Director dr. sarah-marie belcastro (smbelcas) will be joined by MIT graduate student and MathILy Apprentice Instructor Hannah Alpert (snorkack) to lead participants in an exploration of the N-cube from geometric and combinatorial viewpoints. Time will be reserved for a discussion of {MathILy, MathILy-Er} along with any questions you may have about the programs and application process.
kguillet 2015-04-23 19:31:41
For now, please hold your questions -- we'll let you know when you can start asking questions. Also, due to the number of people attending tonight, we may not be able to get to every question.
kguillet 2015-04-23 19:31:52
And now some introductions...
kguillet 2015-04-23 19:31:55
sarah-marie belcastro (smbelcas) earned her Ph.D. in mathematics from the University of Michigan back in 1997 and did her undergraduate work in mathematics and astronomy at Haverford College well before that. She currently directs the summer program MathILy, and has taught a huge variety of mathematics courses---standard and non-standard, undergraduate and graduate in level---to college students and to high-ability high-school students, at institutions including Smith College, Bowdoin College, Sarah Lawrence College, and the Hampshire College Summer Studies in Mathematics. sarah-marie's favorite research is in topological graph theory. Among her many non-pure-mathematics interests are the mathematics of knitting, pharmacokinetics, dance (principally ballet and modern), and changing the world. You may find tons of information (about her, and about other things) at her website http://www.toroidalsnark.net.
kguillet 2015-04-23 19:32:11
Hannah Alpert (snorkack) is a graduate student at MIT, currently studying geometry. She authored/co-authored 6 mathematical research papers before starting graduate school. Hannah has taught at MathPath and at Mathcamp and at the Boston Math Circle, and taught at MathILy 2013 and 2014; she describes her preferred mode of teaching as 'chaos.'
kguillet 2015-04-23 19:32:22
Okay, I'll now hand the room off to your discussion leader for today, sarah-marie!
smbelcas 2015-04-23 19:32:28
Hi, everybody!
smbelcas 2015-04-23 19:32:34
I see there are some students from my past AoPS classes here .
smbelcas 2015-04-23 19:32:46
Welcome to this combined-topic Math Jam! We are going to do some MAAAAAAAATH, and then I'll tell you a tiny bit about {MathILy, MathILy-Er}, and then Hannah and I will answer questions about {MathILy, MathILy-Er}.
smbelcas 2015-04-23 19:33:05
You might like to have some scratch paper handy.
smbelcas 2015-04-23 19:33:16
Those of you who have had me for class know that after I ask a question, I usually won't say anything until you have collectively responded. So don't be surprised if things look quiet for a few moments---we're trying to give you a little bit of thinking/typing space.
smbelcas 2015-04-23 19:33:31
Additionally, please explain your reasoning when you respond. And pay some attention to the responses that get passed into the classroom---they are chosen to help everyone's thinking.
smbelcas 2015-04-23 19:33:45
Finally, the math we do this evening is a tiny sample of what is done at {MathILy, MathILy-Er}. We do this same material faster and we take it much further; today we're just scratching the surface. (If you think it's slow/easy at the start, be patient---it will ramp up!) And of course because {MathILy, MathILy-Er} are face-to-face, class is all much louder and laughier and more student-run.
Wiggle Wam 2015-04-23 19:33:57
How is MAAAAAAAATH different from math ?
smbelcas 2015-04-23 19:34:08
MAAAAAAAAAATH is obviously better.
smbelcas 2015-04-23 19:34:29
And MAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAATH is even better.
smbelcas 2015-04-23 19:34:33
Let's start with a picture:
smbelcas 2015-04-23 19:34:37
//cdn.artofproblemsolving.com/images/6298f39c569c879a22f0eba28f1690ba491938c8.jpg
smbelcas 2015-04-23 19:34:45
What do you think comes next in this sequence? (Just the one thing that comes next...)
beanielove2 2015-04-23 19:35:23
A cube!
goblashapa 2015-04-23 19:35:23
a cube?
johngraham 2015-04-23 19:35:23
cube?
Wiggle Wam 2015-04-23 19:35:23
Cube?
Gwena 2015-04-23 19:35:23
Cube
TurtlePie 2015-04-23 19:35:23
CUBE
smbelcas 2015-04-23 19:35:27
Right, it should be a cube.
smbelcas 2015-04-23 19:35:31
//cdn.artofproblemsolving.com/images/8cbcd6a5974a2324387e6d8fbf6f1b03c099c93d.jpg
smbelcas 2015-04-23 19:35:40
And what do you think comes after that?
TurtlePie 2015-04-23 19:36:30
Hypercube
Gwena 2015-04-23 19:36:30
tesseract
swamih 2015-04-23 19:36:30
a figure in 4d
goblashapa 2015-04-23 19:36:30
maybe a fourth dimension cube or something?
beanielove2 2015-04-23 19:36:30
4-dimensional cube!
Meimeijy 2015-04-23 19:36:30
hypercube
beanielove2 2015-04-23 19:36:30
4-dimensional cube, or a tessarect.
Wiggle Wam 2015-04-23 19:36:30
A fourth dimensional cube
smbelcas 2015-04-23 19:36:32
Now we run into a problem. Who knows what hypercube or tesseract or 4-dimensional cube mean? These terms aren't defined. Let's back up a little bit.
smbelcas 2015-04-23 19:36:46
We understand at a glance that there is a progression here. What exactly is progressing as we go further in the sequence---what's changing?
goblashapa 2015-04-23 19:37:31
the dimension.
johngraham 2015-04-23 19:37:31
the dimentions
Meimeijy 2015-04-23 19:37:31
the dimensions
Silverfang 2015-04-23 19:37:31
Number of dimensions.
Gwena 2015-04-23 19:37:31
dimension
smbelcas 2015-04-23 19:37:36
Indeed, it's the dimension. What are the dimensions of each of those first four objects in the sequence?
TurtlePie 2015-04-23 19:38:20
0, 1, 2, 3...
Gwena 2015-04-23 19:38:20
0, 1, 2, 3
osik 2015-04-23 19:38:20
goblashapa 2015-04-23 19:38:20
0,1,2,3 respectively
Wiggle Wam 2015-04-23 19:38:20
0,1,2,3
smbelcas 2015-04-23 19:38:22
Without getting into the details of the definition of "dimension" (which turns out to need some fairly advanced mathematics to describe precisely), we will agree that a point has 0 dimensions, so these four objects have dimension 0, 1, 2, 3.
smbelcas 2015-04-23 19:38:31
Now, what can we say about the next object in the sequence?
Gwena 2015-04-23 19:39:34
It has dimension 4
goblashapa 2015-04-23 19:39:34
It has four dimensions?
Wiggle Wam 2015-04-23 19:39:34
4 dimensions
kingda1 2015-04-23 19:39:34
Approaching the 4th dimension... *spooky music*
gxah 2015-04-23 19:39:34
it has 4 dimensions
smbelcas 2015-04-23 19:39:37
Right, it should be 4-dimensional. This sets us up for thinking more deeply:
smbelcas 2015-04-23 19:39:41
How do we form this sequence? That is, how do we go from one object in the sequence to the next one?
gxah 2015-04-23 19:40:46
+ 1 dimension
goblashapa 2015-04-23 19:40:46
we ascend the dimensions? dunno.
smbelcas 2015-04-23 19:40:49
Somehow we're going up a dimension. But how, precisely, do we do that?
osik 2015-04-23 19:40:58
TurtlePie 2015-04-23 19:40:58
We add a new dimension perpendicular to the rest
smbelcas 2015-04-23 19:41:42
These are reasonable general descriptions. Let's look at the process one step at a time. What do we do to go from a point to a line segment?
Gwena 2015-04-23 19:42:18
We add another point
goblashapa 2015-04-23 19:42:18
Use another point
kingda1 2015-04-23 19:42:18
add a point beside it
smbelcas 2015-04-23 19:42:27
Yes, but if that's all we do, we just get two points.
wassup 2015-04-23 19:42:31
draw many points next to each other
osik 2015-04-23 19:42:37
smbelcas 2015-04-23 19:42:45
It's hard to do infinitely many things.
Wiggle Wam 2015-04-23 19:42:50
"extend" the point in some sense I guess
goblashapa 2015-04-23 19:43:21
make a string of points or something?
Gwena 2015-04-23 19:43:21
Is there no way to simply add another point and connect it to the existing point?
smbelcas 2015-04-23 19:43:24
Ah! Let's combine the ideas.
smbelcas 2015-04-23 19:43:39
We moosh the point one unit over, and keep the trail it leaves.
smbelcas 2015-04-23 19:43:46
//cdn.artofproblemsolving.com/images/e26570af1191d9d8d9a6b44346581a4b77ba4264.jpg
smbelcas 2015-04-23 19:44:04
What do we do to go from a line segment to a square?
champion999 2015-04-23 19:44:54
we moosh the line upwards
Gwena 2015-04-23 19:44:54
We smooth the line over for the same length as the line
goblashapa 2015-04-23 19:44:54
'moosh' the line one unit over and keep the trail
TurtlePie 2015-04-23 19:44:54
Drag the line
osik 2015-04-23 19:44:54
smbelcas 2015-04-23 19:44:58
//cdn.artofproblemsolving.com/images/94dc80d92afb3badeecb4281212bf6695b3080e7.jpg
smbelcas 2015-04-23 19:45:03
We moosh the segment one unit over, and keep the trail it leaves. But wait---what direction do we moosh in? Does it matter?
Wiggle Wam 2015-04-23 19:46:01
Perpendicular to the line segment
TurtlePie 2015-04-23 19:46:01
Perpendicular direction
smbelcas 2015-04-23 19:46:06
Yes, it matters a lot. We have to moosh perpendicular to the line segment, or else we'll get a parallelogram instead of a square.
smbelcas 2015-04-23 19:46:14
(If you're worried about what moosh really means, it is a type of Cartesian product. Of course, for that to make sense, you have to already know what a Cartesian product is...)
smbelcas 2015-04-23 19:46:29
Okay. What about going from a square to a cube?
champion999 2015-04-23 19:47:22
moosh it upwards!
kingda1 2015-04-23 19:47:22
moosh again
Gwena 2015-04-23 19:47:22
move a square perpendicularly for a specified distance
Wiggle Wam 2015-04-23 19:47:22
Moosh the square up
Philip7086 2015-04-23 19:47:22
moosh iy perpendicular to the plane of the square.
goblashapa 2015-04-23 19:47:22
moosh the square over perpendicular to the square
smbelcas 2015-04-23 19:47:24
Right, we moosh the square one unit in a direction perpendicular to the square, and keep the trail it leaves.
smbelcas 2015-04-23 19:47:29
//cdn.artofproblemsolving.com/images/bd3afe6f3736e5eab141129dec60c17c8ea208db.jpg
smbelcas 2015-04-23 19:47:35
And what comes next?
champion999 2015-04-23 19:48:54
I have noticed that a segment is connecting two parallel points, a square connects two parallel lines, and a cube connects two parallel squares. Maybe a tesseract connects two parallel cubes?
osik 2015-04-23 19:48:54
TurtlePie 2015-04-23 19:48:54
Moosh the cube in a new perpendicular dimension
kingda1 2015-04-23 19:48:54
mush... On all sides??? Diagonal out leaving an outside cube and an inside one?
smbelcas 2015-04-23 19:48:58
Exactly. We moosh the cube one unit in a direction perpendicular to the cube, and keep the trail it leaves.
Gwena 2015-04-23 19:49:08
We must move the cube perpendicularly in relation to itself
smbelcas 2015-04-23 19:49:16
Yes.
smbelcas 2015-04-23 19:49:30
//cdn.artofproblemsolving.com/images/70d3b7691aa6e0910a209a8604067c6084485c20.jpg
smbelcas 2015-04-23 19:49:48
We have agreed that the resulting object is 4-dimensional, and it's certainly cube-like, so let's call it a 4-cube.
smbelcas 2015-04-23 19:49:57
Here are all the cubes we've made so far, for reference:
smbelcas 2015-04-23 19:50:03
//cdn.artofproblemsolving.com/images/b8fe693a72a95f2a67f5e504f277414836e01a35.jpg
smbelcas 2015-04-23 19:50:28
How do we make a 5-cube? Can you draw one (on paper, not on screen---that would take too long)?
thkim1011 2015-04-23 19:51:42
you can do the same thing in some other direction
csmath 2015-04-23 19:51:42
You take 4 cubes and move them perpendicularly into the 5th dim
smbelcas 2015-04-23 19:51:49
Just as we have done for earlier dimensions, we moosh the 4-cube one unit in a direction perpendicular to the 4-cube, and keep the trail it leaves.
mathtastic 2015-04-23 19:52:10
where is this going
smbelcas 2015-04-23 19:52:12
How do we make an $n$-cube?
Wiggle Wam 2015-04-23 19:53:16
Moosh the n-1 cubes
csmath 2015-04-23 19:53:16
take n-1 cube and move it perpendicularly into the nth dimension
champion999 2015-04-23 19:53:16
we keep mooshing the cubes in a direction perpendicular to it.
TurtlePie 2015-04-23 19:53:16
Moosh $n$ times in $n$ perpendicular directions
swamih 2015-04-23 19:53:16
take $n-1$ cubes and move them perpendicularly to the $n$th dimension
Gwena 2015-04-23 19:53:16
We move an (n-1) cube perpendicularly by a specified unit from the (n-1) cube to form an n-cube
smbelcas 2015-04-23 19:53:24
Yup. We moosh a $(n-1)$-cube one unit in a direction perpendicular to the $(n-1)$-cube, and keep the trail it leaves.
smbelcas 2015-04-23 19:53:41
Some of you have wondered where these extra perpendicular directions are.
smbelcas 2015-04-23 19:53:59
One answer: Usually when you graph things, you use the $x$ axis and the $y$ axis, and for 3-dimensional things, the $z$ axis. Here, we also use the $w$ axis, and for the 5-cube, the $v$ axis.
smbelcas 2015-04-23 19:54:16
The $w$ axis is perpendicular to the $x, y$, and $z$ axes. The $v$ axis is perpendicular to the $x, y, z$, and $w$ axes.
csmath 2015-04-23 19:54:26
In higher dimensions of course
smbelcas 2015-04-23 19:54:29
Just like you can't draw the $z$ axis as perpendicular to the $x$ and $y$ axes on paper, but you can still understand where it goes in 3 dimensions...
smbelcas 2015-04-23 19:54:40
...you can't place the $w$ axis as perpendicular to the $x, y$, and $z$ axes in regular space, but you can still understand where it goes in 4 dimensions. There isn't room for the $w$ axis in regular space, but there is enough room in your head. With practice, you can visualize it pretty clearly.
Wiggle Wam 2015-04-23 19:54:59
We can visualize 3 dimensions by drawing in two dimensions (i.e. we can draw a picture of a cube on paper). But I'm just not seeing 4 dimensions on this two dimensional screen. Would it be possible to visualize 4 dimensions by making a 3D construction?
smbelcas 2015-04-23 19:55:12
Yes, it is.
smbelcas 2015-04-23 19:55:42
Historically, that was one of the most popular ways to do it.
smbelcas 2015-04-23 19:55:56
And yes, there was a while when visualizing the fourth dimension was all the rage!
Gwena 2015-04-23 19:56:03
Are you just out of luck after 4 dimensions, then?
smbelcas 2015-04-23 19:56:13
...No, but it takes a lot more practice to visualize.
smbelcas 2015-04-23 19:56:19
Anyway: 5-cube!
smbelcas 2015-04-23 19:56:22
//cdn.artofproblemsolving.com/images/dd763f26ffc54dc31675c14ab24855ed13ae102b.jpg
smbelcas 2015-04-23 19:56:30
6-cube!
smbelcas 2015-04-23 19:56:34
//cdn.artofproblemsolving.com/images/6ef97b464262f8a41f17907a82fecec00f700fb1.jpg
hzbest 2015-04-23 19:56:50
Woah
Gwena 2015-04-23 19:56:50
That's beautiful.
csmath 2015-04-23 19:56:54
This is highly intimidating-looking.
SirCalcsALot 2015-04-23 19:56:57
Really complicated!
smbelcas 2015-04-23 19:57:04
Okay, that was just for fun. Sometimes I get a bit excited.
azmath333 2015-04-23 19:57:08
7-cube?
smbelcas 2015-04-23 19:57:19
No, because at that point it gets too messy.
smbelcas 2015-04-23 19:57:30
So far we have only constructed the $n$-cube visually. Let us now situate it in space. Again, we'll go dimension by dimension.
smbelcas 2015-04-23 19:57:42
The most convenient place to put a single point (a 0-cube) is at 0.
smbelcas 2015-04-23 19:57:58
What should the ends of our 1-cube line segment be, in terms of coordinates?
Gwena 2015-04-23 19:58:57
0 and 1?
somepersonoverhere 2015-04-23 19:58:57
(0) and (1)
SirCalcsALot 2015-04-23 19:58:57
$0$ and $1$
Hydroxide 2015-04-23 19:58:57
0 and 1
smbelcas 2015-04-23 19:58:59
Yes, we put them at 0 and at 1.
smbelcas 2015-04-23 19:59:10
What about the corners of a square (a 2-cube)?
Gwena 2015-04-23 19:59:44
(0,0) (1,0) (0,1) (1,1)
thkim1011 2015-04-23 19:59:44
(0,0), (0,1), (1,0), (1,1)
SirCalcsALot 2015-04-23 19:59:44
(0,0), (0,1), (1,1), (1,0)
goblashapa 2015-04-23 19:59:44
(0,0), (1,0), (0,1), (1,1)
somepersonoverhere 2015-04-23 19:59:44
(0, 0), (1, 0), (0, 1), (1, 1)
champion999 2015-04-23 19:59:47
(0,0) (0,1) (1,0) (1,1)
smbelcas 2015-04-23 19:59:49
For consistency, we place them at (0,0), (0,1), (1,0), and (1,1).
smbelcas 2015-04-23 19:59:59
Now I'm just going to ask you a volley of questions: How many corners does a 3-cube have? What about a 4-cube? ...an $n$-cube? What are the coordinates of those corners? Is there an easy way to describe them?
smbelcas 2015-04-23 20:00:58
I'm going to let the responses pile up for a bit, and then pass your responses through in batches by common theme.
Gwena 2015-04-23 20:02:33
2^3, 2^4, 2^5, ... 2^n
somepersonoverhere 2015-04-23 20:02:33
3-cube has 8 corners, 4-cube has 16 corners, n-cube has ;$2^n$ corners
Philip7086 2015-04-23 20:02:33
8, 16, 2^n ?
Darn 2015-04-23 20:02:33
A 3-d cube has 8, a 4-d cube has 16, and generally an n-cube has $2^n$
smbelcas 2015-04-23 20:02:36
A 3-cube has 8 corners. A 4-cube has 16 corners.
Darn 2015-04-23 20:03:02
2^n
champion999 2015-04-23 20:03:02
2^n
TurtlePie 2015-04-23 20:03:02
$2^n$
Wiggle Wam 2015-04-23 20:03:02
An n-cube has $2^n$ corners.
azmath333 2015-04-23 20:03:02
$2^n$
ImpossibleCube 2015-04-23 20:03:02
2^n corners for an n cube?
goblashapa 2015-04-23 20:03:02
number of corners: 2^n
Gwena 2015-04-23 20:03:02
The number of points in an n-cube is 2^n
smbelcas 2015-04-23 20:03:05
There are lots of conjectures that an $n$-cube has $2^n$ corners.
Darn 2015-04-23 20:03:31
So like, for a 3-d cube, we have 000,001,010,011,100,101,110,111 as the corners which correspond to $(0,0,0),(0,0,1),(0,1,0),\ldots,(1,1,1)$.
smbelcas 2015-04-23 20:03:33
A 3-cube has corners at (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), and (1,1,1).
Darn 2015-04-23 20:03:51
They can be expressed as binary numbers of length n for a n-dimension cube
azmath333 2015-04-23 20:03:51
All the possible binary strings of digits for $n$ digits
Wiggle Wam 2015-04-23 20:03:51
And their vertices can be described by all the different ways you can put a 0 or a 1 into each spot of a set of n spots representing n coordinates
somepersonoverhere 2015-04-23 20:03:51
coordinates are the set of ;$(a_1, a_2,....a_n)$ such that ;$(a_i)$ is 0 or 1
Gwena 2015-04-23 20:03:51
We could describe these points as for an n-cube as for n coordinates having 2 choices 0 or 1
C-bass 2015-04-23 20:03:51
(a,b,c), where a,b, and c are either 0 or 1
ImpossibleCube 2015-04-23 20:03:51
the coordinates for an n-cube are the coordinates in the form (x_1,x_2,x_3...x_n) such that x_i is 0 or 1
smbelcas 2015-04-23 20:03:58
People think that the corners of an $n$-cube can be described as all $n$-tuples with entries that are 0 or 1. But this is a conjecture...
smbelcas 2015-04-23 20:04:11
...and so here are more questions. Can you prove that an $n$-cube has $2^n$ corners? Are you sure that every corner of an $n$-cube should have coordinates with entries that are 0 or 1; why? Does every $n$-tuple with 0/1 entries represent a corner of an $n$-cube?
smbelcas 2015-04-23 20:04:39
These questions are much tougher, but here is where we are really getting to understand the $n$-cube. Let's answer them one at a time.
smbelcas 2015-04-23 20:05:01
Prove that an $n$-cube has $2^n$ corners.
TurtlePie 2015-04-23 20:06:29
You are doubling every previous shape, so you are doubling it, or multiplying by 2
smbelcas 2015-04-23 20:06:34
When we make an $n$-cube from an $(n-1)$-cube, we moosh that $(n-1)$-cube by one unit. There are the "starting" corners and the "ending" corners, so there are twice as many corners in an $n$-cube as in an $(n-1)$-cube.
Wiggle Wam 2015-04-23 20:06:40
When you "moosh", you add another set of corners identical to the first; in other words, you double the number of corners.
Gwena 2015-04-23 20:06:56
Each time we create a new n-cube, we are moving the previous cube to a new location while keeping the old one. In effect this doubles the number of points
thkim1011 2015-04-23 20:07:08
by a simple induction argument $a_n = 2^n$ satisfies a_n = 2 a_(n-1) with a_0 = 1.
smbelcas 2015-04-23 20:07:12
That's not enough to say that the number of corners is $2^n$, though. We also have to remember that we can manually count to see that the number of corners of the {0-cube, 1-cube, 2-cube, 3-cube} is {1, 2, 4, 8} so if we continue to double we'll always get $2^n$.
smbelcas 2015-04-23 20:07:29
Are you sure that every corner of an $n$-cube should have coordinates with entries that are 0 or 1; why?
Gwena 2015-04-23 20:09:20
We define a unit as length 1. Each time we move by 1, expanding into a new dimension. We either remain in the same location (0) or move (1). We do not have any other options.
smbelcas 2015-04-23 20:09:38
The corners of an $n$-cube must have coordinate entries that are 0 or 1 if we situate our original point at 0, because every corner is either a "starting" corner (0 entry) or an "ending" corner (1 entry) in the last coordinate, and has the 0/1 entries from lower-dimensional cubes in the first $n-1$ coordinates.
goblashapa 2015-04-23 20:09:45
because the sides will always be one unit in length and the sides will always be along an axis since the cube is in the same dimension as the number of dimensions it has?
smbelcas 2015-04-23 20:10:36
Does every $n$-tuple with 0/1 entries represent a corner of an $n$-cube?
TurtlePie 2015-04-23 20:11:17
YES
smbelcas 2015-04-23 20:11:22
Explain why!
Wiggle Wam 2015-04-23 20:11:51
Yes, because there are $2^n$ corners in an n-cube and $2^n$ different n-tuples. Also, two corners of an n-cube can't be the same n-tuple.
smbelcas 2015-04-23 20:11:59
One way we can say that every $n$-tuple with 0/1 entries represents a corner of an $n$-cube is by using our previous two arguments: There are $2^n$ $n$-tuples with 0/1 entries, and every corner must be among them. However, there are also $2^n$ corners, so each of those $n$-tuples represents a corner!
CInfinitesimal 2015-04-23 20:13:13
Each of the coordinates have to have the distance 0 or 1 from an axis because it's a unit cube!
Gwena 2015-04-23 20:13:13
If we have each n-cube have options 0 or 1 for its coordinate set and 2^n distinct points, each coordinate set in terms 0 and 1 alone must be a corner of a n-cube.
smbelcas 2015-04-23 20:13:19
Now let's count parts of $n$-cubes: Please help to fill in the table:
smbelcas 2015-04-23 20:13:23
$$\begin{array}{c||c|c|c|c|c|c|c}
{\rm dim} &0&1&2&3&4&5&\dots\\ \hline
\hline
{\rm points} & 1 &2&&& &&\dots\\
\hline
{\rm lines} &0 &1 & & & & &\dots\\
\hline
{\rm squares} &0 &0 & & & & &\dots\\
\hline
{\rm 3-cubes} &0 &\hspace{1cm} & & & & &\dots\\
\hline
{\rm 4-cubes}&\hspace{1cm} &0 &\hspace{1cm}&\hspace{1cm}&\hspace{1cm}&\hspace{1cm}&\dots\\
\hline
\end{array}$$
smbelcas 2015-04-23 20:13:31
So, for example, you can say "points in 2-cube is 4" to fill in the next entry of the "points" row.
smbelcas 2015-04-23 20:14:27
I'll pass the answers through, column by column.
smbelcas 2015-04-23 20:15:21
For 2-cubes:
Wiggle Wam 2015-04-23 20:15:26
4, 4,1
mathwhiz918 2015-04-23 20:15:26
4 points in 2-cube
Gwena 2015-04-23 20:15:26
dim 2, points is 4
mathwhiz918 2015-04-23 20:15:30
4 lines in 2-cube
mathwhiz918 2015-04-23 20:15:30
one square in 2-cube
smbelcas 2015-04-23 20:15:44
What about 3-cubes or 4-cubes?
goblashapa 2015-04-23 20:16:09
0 and 0
mathwhiz918 2015-04-23 20:16:48
0 3-cubes and 4-cubes for dim 0 and 1
smbelcas 2015-04-23 20:16:55
Also true.
smbelcas 2015-04-23 20:17:00
Let's do the 3-cube column:
Wiggle Wam 2015-04-23 20:17:11
8, 12, 6, 1
goblashapa 2015-04-23 20:17:14
8 12 6 1 0
smbelcas 2015-04-23 20:17:55
What should the numbers be for the 4-cube?
smbelcas 2015-04-23 20:19:04
(This is why I suggested at the beginning that scratch paper might be useful!)
Gwena 2015-04-23 20:19:17
16
goblashapa 2015-04-23 20:19:17
16 points 32 lines
goblashapa 2015-04-23 20:20:16
i know its 1 4 cube
smbelcas 2015-04-23 20:21:08
Anyone have any numbers for the 5-cube?
goblashapa 2015-04-23 20:21:43
32 points
Darn 2015-04-23 20:21:43
32, 80, 80, 40, 10, 1?
smbelcas 2015-04-23 20:21:53
Wait a minute. Where are those numbers coming from? Did you draw a 5-cube earlier, and have been counting from it?
Darn 2015-04-23 20:22:10
um guessing
smbelcas 2015-04-23 20:22:17
I'm just not sure of all those numbers. Well, here is the updated table:
smbelcas 2015-04-23 20:22:28
(And I'm going to give you the correct 4-cube numbers...)
smbelcas 2015-04-23 20:22:30
$$\begin{array}{c||c|c|c|c|c|c|c}
{\rm dim} &0&1&2&3&4&5&\dots\\ \hline
\hline
{\rm points} & 1 &2&4&8&16&32&\dots\\
\hline
{\rm lines} &0 &1 &4 &12 &32 & &\dots\\
\hline
{\rm squares} &0 &0 &1 &6 &24 & &\dots\\
\hline
{\rm 3-cubes} &0 &0 &0 &1 &8 & &\dots\\
\hline
{\rm 4-cubes}&0 &0 &0 &0 &1 & &\dots\\ \hline
&\hspace{1cm} &\hspace{1cm}&\hspace{1cm}&\hspace{1cm}&\hspace{1cm}&\hspace{1cm}& \\
\end{array}$$
smbelcas 2015-04-23 20:22:42
The 5-cube numbers are mostly left out because I'm just not convinced yet.
smbelcas 2015-04-23 20:22:50
Let me give you some notation: Let $C_n$ denote the $n$-cube.
smbelcas 2015-04-23 20:23:05
And let $f_k(C_n) = $ the number of $k$-cubes in $C_n$. (The $f$ stands for $f$aces.)
smbelcas 2015-04-23 20:23:27
We already proved that $f_0(C_n) = 2^n$. Do you have any conjectures based on the data we have in the table?
CInfinitesimal 2015-04-23 20:23:46
point=2^n, line (dim n)=2*line(dim n-1)+2^(n-1)
Darn 2015-04-23 20:23:46
Well lines is like $n\cdot 2^{n-1}$ i think
Gwena 2015-04-23 20:23:52
Well the ones will continue diagonally
Gwena 2015-04-23 20:24:46
And so will the 0s
CInfinitesimal 2015-04-23 20:25:48
face (dim n)=2 face(n-1) + line (dim n-1) so the function is 2(# of figures in C(n-1))+# of figures from a previous dimension (C n-1)
smbelcas 2015-04-23 20:26:35
Any conjectures in general dimension $n$?
smbelcas 2015-04-23 20:27:12
Or about general face dimensions?
Gwena 2015-04-23 20:27:18
Dimension n will have only one n-cube
Darn 2015-04-23 20:27:23
ooh squares looks like $\dbinom{n}{2}\cdot 2^{n-2}$
Darn 2015-04-23 20:28:12
hmm $\dbinom{n}{k}\cdot 2^{n-k}$ where $k$ denotes the dimension of the value asked?
smbelcas 2015-04-23 20:28:40
I would like to see a generalization of CInfinitesimal's conjecture as well.
Gwena 2015-04-23 20:29:50
I'm not sure I understand CIninitesimal's conjecture.
smbelcas 2015-04-23 20:30:09
Here's something CInfinitesimal said earlier about a specific case:
CInfinitesimal 2015-04-23 20:30:12
Because each time you expand another dimension you double the existing lines and connect the pairs of corresponding points
CInfinitesimal 2015-04-23 20:30:34
2(# of figures in C(n-1))+# of figures from a previous dimension (C n-1) Because each time the dim expands everything doubles and the figures from a previous dimension are paired up and connected
smbelcas 2015-04-23 20:30:45
Okay! Now we're cookin'!
smbelcas 2015-04-23 20:30:50
Conjecture 1: $f_k(C_n) = 2f_k(C_{n-1})+f_{k-1}(C_{n-1})$.
smbelcas 2015-04-23 20:31:04
(I'm using the notation I introduced a bit ago.)
smbelcas 2015-04-23 20:31:09
Conjecture 2: $f_k(C_n) = {n\choose k}2^{n-k}$. (This one only makes sense if you know binomial coefficients already.)
smbelcas 2015-04-23 20:31:30
Can you prove either of these?
smbelcas 2015-04-23 20:32:24
Conjecture 1 follows from our construction of the $n$-cube. How does that work, exactly?
Gwena 2015-04-23 20:33:12
We mosh an n-1 cube perpendicularly to form a n cube
CInfinitesimal 2015-04-23 20:33:59
By connected I mean points connected by lines, lines connected by faces, and faces connected by 3-cubes
goblashapa 2015-04-23 20:34:08
We have twice the number of k cubes from Cn-1 because we mooshed it, and then plus the number of k-1 cubes, because we used them to make the trail.
smbelcas 2015-04-23 20:35:14
Let's go through the details. What happens to a corner point when we moosh?
Gwena 2015-04-23 20:36:23
One stays at 0 in the dimension and the other moves to 1
goblashapa 2015-04-23 20:36:23
it gets duplicated and we record the trail
goblashapa 2015-04-23 20:36:23
moved one unit over and keep the trail
Wiggle Wam 2015-04-23 20:36:23
It gets translated to form another corner point.
smbelcas 2015-04-23 20:36:46
It turns into a line segment. Basically it has the "starting" (or 0) point, and the "ending" (or 1) point, and the trail left is a line segment.
smbelcas 2015-04-23 20:36:57
What happens to a line segment when we moosh?
Wiggle Wam 2015-04-23 20:37:40
forms a square
goblashapa 2015-04-23 20:37:40
It gets moved over one unit and we keep the trail to form a 2 cube
smbelcas 2015-04-23 20:37:44
It turns into a square. Basically it has the "starting" (or 0) segment, and the "ending" (or 1) segment, and the trail left is a square.
smbelcas 2015-04-23 20:37:47
What happens to a square when we moosh?
goblashapa 2015-04-23 20:38:22
it becomes a 3 cube
Gwena 2015-04-23 20:38:22
We get a 3-cube
Wiggle Wam 2015-04-23 20:38:22
Forms a cube!!
smbelcas 2015-04-23 20:38:28
It turns into a 3-cube. Basically it has the "starting" (or 0) square, and the "ending" (or 1) square, and the trail left is a 3-cube.
Gwena 2015-04-23 20:38:36
We moosh creating a 3-cube with starting and ending points 0, 1 in the new dimension
smbelcas 2015-04-23 20:38:40
So, more generally, what happens to a $k$-cube when we moosh?
goblashapa 2015-04-23 20:39:43
it becomes a k+1 cube
Wiggle Wam 2015-04-23 20:39:43
Forms a k+1 cube.
Gwena 2015-04-23 20:39:43
A k-cube mooches into a k+1 cube with starting and ending points 0,1 in dimension k+1
smbelcas 2015-04-23 20:39:46
It turns into a $(k+1)$-cube. Basically it has the "starting" (or 0) $k$-cube, and the "ending" (or 1) $k$-cube, and the trail left is a $(k+1)$-cube.
smbelcas 2015-04-23 20:39:53
How does that help us prove Conjecture 1?
Wiggle Wam 2015-04-23 20:41:17
So, when counting the number of k cubes in the n cube, we know that we can form some of these k cubes by mooshing all the k-1 cubes in the n-1 cube.
smbelcas 2015-04-23 20:41:40
The formula says the number of $k$-cubes in an $n$-cube is the same as 2(the number of $k$-cubes in an $(n-1)$-cube) + (the number of $(k-1)$-cubes in an $(n-1)$-cube).
CInfinitesimal 2015-04-23 20:41:58
So each part of an n-cube gets copied in the direction of the n+1 dim by one and the trail left by the part forms a figure in the following dimension of the part itself
Wiggle Wam 2015-04-23 20:42:56
Each k-cube in the n-1 cube gets copied.
TheKid2 2015-04-23 20:42:56
each oart of an n- cube gets copied
smbelcas 2015-04-23 20:43:01
The 2(the number of $k$-cubes in an $(n-1)$-cube) counts the "starting" and "ending" $k$-cubes from a moosh.
swamih 2015-04-23 20:44:03
$2f_kC_{n-1}$
smbelcas 2015-04-23 20:44:12
Yes!
Gwena 2015-04-23 20:44:19
And each k-1 cube creates a new k cube
smbelcas 2015-04-23 20:44:41
The (the number of $(k-1)$-cubes in an $(n-1)$-cube) counts the new $k$-cubes from the moosh, which all came from the trails of $(k-1)$-cubes.
smbelcas 2015-04-23 20:45:03
You've collectively proved Conjecture 1!
smbelcas 2015-04-23 20:45:09
Conjecture 2 is tougher.
smbelcas 2015-04-23 20:45:27
Any ideas?
CInfinitesimal 2015-04-23 20:46:17
n choose k and the power of 2 makes me think of coordinates
smbelcas 2015-04-23 20:47:49
Here we go:
Darn 2015-04-23 20:47:52
Sketch of second generalization? : as we have proven before, there are $2^n$ points. Exactly $2^{k}$ of these points are required to make a cube of dimension $k$. Notice that we must have $k$ distinct dimensions in which to build the cube of dimension $k$ from the n-cube in order to construct such a k-cube. There are $\dbinom{n}{k}$ ways to do this. We then choose the points that we use to make this k-cube out of the total number of such points, which is equal to $\frac{2^{n}}{2^{k}}=2^{n-k}$, a
Darn 2015-04-23 20:47:56
oops my text got cut off luckily i have it on my clipboard: and our result is $\dbinom{n}{k}\cdot2^{n-k}$. (sorry if this is wrong/flawed or is unclear D:)
smbelcas 2015-04-23 20:48:51
What do you all think?
smbelcas 2015-04-23 20:49:33
(I told you this one was tougher!)
smbelcas 2015-04-23 20:50:06
Let me rephrase Darn's argument a little bit.
smbelcas 2015-04-23 20:50:12
At every corner of an $n$-cube, there are $n$ edges because we have done $n$ mooshes. Every $k$ of these determine a $k$-cube. This is true for every one of the $2^n$ corners. However, this overcounts by the number of corners in a $k$-cube, which is $2^k$. Thus the total number of $k$-cubes is ${n\choose k}\dfrac{2^n}{2^k} = {n\choose k}2^{n-k}$.
smbelcas 2015-04-23 20:50:48
This isn't quite an airtight argument, though---how do we know that every $k$ edges determine a $k$-cube?
Darn 2015-04-23 20:53:26
You are fixing the dimensions, not the lines. There exists a configuration with those specified dimensions and points such that it makes a k-cube.
smbelcas 2015-04-23 20:53:46
To make this rigorous, we can use Cartesian products or the coordinate description of the $n$-cube to justify this.
smbelcas 2015-04-23 20:54:05
Okay, let's switch gears and talk about {MathILy, MathILy-Er} so that there's time for you to ask lots of questions. (I hope you enjoyed the math!)
smbelcas 2015-04-23 20:54:16
{MathILy, MathILy-Er} are intensive residential summer programs for mathematically excellent secondary students.
smbelcas 2015-04-23 20:54:26
As we say on the website (http://www.mathily.org), {MathILy, MathILy-Er} focus on participants exploring and creating mathematics. Instructors provide the framework and you get to make (and prove!) the conjectures. You will encounter new ideas, improve your problem-solving skills, learn lots and lots of advanced mathematics, and hone your overall thinking skills. You'll meet others like you. (Yes, really. We promise.) Most of all, you will find serious mathematics infused with levity.
{MathILy, MathILy-Er} are five weeks of maximized mathematical marvelousness.
smbelcas 2015-04-23 20:54:44
The programs share an application process---you apply to both programs at once. You take an Exam Assessing Readiness and fill out some information on the Short Form and Not-as-Short Form. Based on these things (and comments from a recommender) the {MathILy, MathILy-Er} Directors decide whether you are qualified, and if so, for which program. MathILy-Er is designed for students who are a little bit earlier in their mathematical development than MathILy students.
smbelcas 2015-04-23 20:54:57
Please ask questions!
goblashapa 2015-04-23 20:55:49
Can we apply if we are currently in 8th grade for this summer?
smbelcas 2015-04-23 20:56:13
Yes. We have had several applications from 8th graders. It's not as easy to get in, but please do try!
Wiggle Wam 2015-04-23 20:56:18
What quite do you mean by "a little bit earlier"? Like, is MathILy-Er for more middle schoolers while MathILy is geared towards high school?
smbelcas 2015-04-23 20:56:56
No---both are geared toward high school. But MathILy-Er is for younger students within high school, and high-school students who aren't quite ready for MathILy (because it's really really intense).
Darn 2015-04-23 20:57:02
where is this located
smbelcas 2015-04-23 20:57:20
This summer, MathILy is near Philadelphia and MathILy-Er is near Portland.
Philip7086 2015-04-23 20:57:27
what does ILy stand for?
smbelcas 2015-04-23 20:57:33
Infused with Levity.
CInfinitesimal 2015-04-23 20:57:51
How much calc does MathILy involve?
smbelcas 2015-04-23 20:57:54
None!
Wiggle Wam 2015-04-23 20:58:01
But the EAR would be due tomorrow! Could we still possibly get in?
smbelcas 2015-04-23 20:58:10
Uh... no, the EAR is not due tomorrow.
smbelcas 2015-04-23 20:58:32
We have a deadline of April 27th for full consideration of applications---that's Monday.
smbelcas 2015-04-23 20:58:58
But there may be spots left, and we will certainly start a waiting list if not, so later applications are fine too (at least for a while).
goblashapa 2015-04-23 20:59:06
What is the EAR
smbelcas 2015-04-23 20:59:12
Exam Assessing Readiness!
swamih 2015-04-23 20:59:24
Where can we find the EAR?
smbelcas 2015-04-23 20:59:41
You can't. You have to ask for one personally.
beanielove2 2015-04-23 21:00:13
How much does the MathILy, MathILy-Er camp cost?
hzbest 2015-04-23 21:00:13
What's the tuition?
smbelcas 2015-04-23 21:00:27
$4600 for the five weeks, and there's need-based financial aid available.
TheKid2 2015-04-23 21:00:33
How do you ask for EAR?
smbelcas 2015-04-23 21:00:46
Submit a Short Form on the website.
goblashapa 2015-04-23 21:00:55
How hard is it to get in? like how many people apply and how many get it. Also, what does the test ask, like what level?
smbelcas 2015-04-23 21:01:13
Let's see. There are lots of ways to answer this. Maybe Hannah and I will both answer
smbelcas 2015-04-23 21:01:40
It's hard to get in. We certainly take fewer than 50% of applicants, including both programs.
smbelcas 2015-04-23 21:02:05
MathILy itself might be more like 25%. It changes every year.
smbelcas 2015-04-23 21:02:35
How many applicants is different every year, and lots of people only do part of the application, so it's hard to measure. I think a lot of people start applications and then give up on the EAR.
smbelcas 2015-04-23 21:03:03
I have no idea how to describe the level of the EAR.
Darn 2015-04-23 21:03:15
um does the acronym ily mean "i love you?"
smbelcas 2015-04-23 21:03:29
People often ask this: No, but we're happy to have you think it
snorkack 2015-04-23 21:04:08
Our students are at a similar level to those at other national summer programs. The EAR helps determine both (a) whether you know enough math that you'd be able to understand what's going on during the program; and (b) whether you approach math with an attitude that would enable you to grow a lot during the program. It's hard to compare to other tests.
smbelcas 2015-04-23 21:04:57
And also whether you can keep up with the program---we certainly have applicants who clearly could understand the math given enough time.
smbelcas 2015-04-23 21:05:14
But lots of applicants would get snowed under by the pace, and they are not admitted.
TheMaskedMagician 2015-04-23 21:05:21
what subjects does mathILy cover?
smbelcas 2015-04-23 21:05:44
We focus on discrete math, and the core curriculum is combinatorics, graph theory, and theoretical linear algebra.
smbelcas 2015-04-23 21:05:56
But then we have a zillion topics during Week of Chaos.
smbelcas 2015-04-23 21:06:13
And 2-week long focused classes after that.
beanielove2 2015-04-23 21:06:24
What's Week of Chaos
smbelcas 2015-04-23 21:06:43
It's.... a week... full of chaos.
goblashapa 2015-04-23 21:06:45
Is lodging provided or where do we stay?
snorkack 2015-04-23 21:07:18
Students and staff all live together in a college dorm. We eat together at the college cafeteria.
CInfinitesimal 2015-04-23 21:07:41
Chaos theory? Interested...
smbelcas 2015-04-23 21:08:00
Chaos theory is not always a topic. But it was a topic last summer!
goblashapa 2015-04-23 21:09:01
Will you be one of the teachers?
smbelcas 2015-04-23 21:09:09
Yes.
goblashapa 2015-04-23 21:09:28
At mathily or mathilyer
smbelcas 2015-04-23 21:09:37
Hannah and I are both teaching at MathILy this summer.
smbelcas 2015-04-23 21:10:11
My former student Jonah, who is approximately 46 times as fun as I am, is teaching at MathILy-Er.
snorkack 2015-04-23 21:11:12
Aspects of Week of Chaos that are chaotic: class topics are nominated by students; students have classes on 5 different topics per day; the schedule is made a day or two before the beginning of the week and involves many index cards and the help of a computer; I once taught a class session where everything anyone said was in song.
smbelcas 2015-04-23 21:11:43
Oh, yeah, and I taught a class session where I didn't speak.
snorkack 2015-04-23 21:12:14
Was that the same class section where all you did was erase things that students wrote?
smbelcas 2015-04-23 21:12:35
Um, yes, I think so. But that happens sort of regularly, actually.
smbelcas 2015-04-23 21:12:56
I mean, not every day. But at least once per year.
smbelcas 2015-04-23 21:13:05
Or maybe once/week.
goblashapa 2015-04-23 21:13:09
Wait so you guys will send us a EAR test and we have all the time we want before the due date to complete it?
smbelcas 2015-04-23 21:13:14
Sort of.
smbelcas 2015-04-23 21:13:28
You should only spend 2--4 hours on it. Most people take all 4 and don't finish all the problems.
smbelcas 2015-04-23 21:13:41
But you can pick the hours you spend on it.
smbelcas 2015-04-23 21:14:02
And it's the Minion who sends you the EAR.
goblashapa 2015-04-23 21:14:10
Is it online?
smbelcas 2015-04-23 21:14:40
The Minion emails you a PDF. You write on it and scan it and send it to me. Or put it in the postal mail, if you don't have access to a scanner.
smbelcas 2015-04-23 21:14:59
Okay, most people print out the EAR before writing on it.
goblashapa 2015-04-23 21:15:01
If we fill out the short form, are we guaranteed to receive the EAR?
smbelcas 2015-04-23 21:15:16
Yup. The Minion sends them out a couple of times per day.
snorkack 2015-04-23 21:15:41
(Assuming that the email address you put down is actually yours.)
goblashapa 2015-04-23 21:16:19
in the short form it asks for the math class you are taking would an AOPS class count
smbelcas 2015-04-23 21:16:38
Only if it's a curricular class (like Precalculus).
smbelcas 2015-04-23 21:17:23
Plenty of people say things like "9th grade Trigonometry" or "homeschooled Algebra II"
goblashapa 2015-04-23 21:17:35
Wait do we need something like a teacher recommendation form or of the likeness?
smbelcas 2015-04-23 21:17:50
The Not-as-Short Form asks for the name/email of a recommender.
smbelcas 2015-04-23 21:17:59
Submitting that form auto-emails the recommender with instructions.
smbelcas 2015-04-23 21:20:07
It's getting late---any last questions before we wrap up this MAAAAAAAAAAATH Jam?
smbelcas 2015-04-23 21:20:59
Hurray! We have answered all possible questions! (Okay, not really.)
kguillet 2015-04-23 21:21:20
Alright everyone, that wraps things up for today’s Math Jam. Thank you for coming, and a special thanks to sarah-marie and Hannah for the great discussion!
kguillet 2015-04-23 21:21:35
This room will be closing in a couple of minutes.

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