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

Go back to the Math Jam ArchiveAoPS 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).

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.

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.

Hi everyone. We'll get started in just a couple of minutes.

kguillet
2015-04-23 19:30:55

Alright, let's get going!

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!

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.

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.

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.

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.

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.

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...

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.

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.'

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!

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!

Hi, everybody!

smbelcas
2015-04-23 19:32:34

I see there are some students from my past AoPS classes here .

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}.

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.

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.

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

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.

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 ?

How is MAAAAAAAATH different from math ?

smbelcas
2015-04-23 19:34:08

MAAAAAAAAAATH is obviously

MAAAAAAAAAATH is obviously

*better*.
smbelcas
2015-04-23 19:34:29

And MAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAATH is

And MAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAATH is

*even better*.
smbelcas
2015-04-23 19:34:33

Let's start with a picture:

Let's start with a picture:

smbelcas
2015-04-23 19:34:37

smbelcas
2015-04-23 19:34:45

What do you think comes next in this sequence? (Just the

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!

A cube!

goblashapa
2015-04-23 19:35:23

a cube?

a cube?

johngraham
2015-04-23 19:35:23

cube?

cube?

Wiggle Wam
2015-04-23 19:35:23

Cube?

Cube?

Gwena
2015-04-23 19:35:23

Cube

Cube

TurtlePie
2015-04-23 19:35:23

CUBE

CUBE

smbelcas
2015-04-23 19:35:27

Right, it should be a cube.

Right, it should be a cube.

smbelcas
2015-04-23 19:35:31

smbelcas
2015-04-23 19:35:40

And what do you think comes after that?

And what do you think comes after that?

TurtlePie
2015-04-23 19:36:30

Hypercube

Hypercube

Gwena
2015-04-23 19:36:30

tesseract

tesseract

swamih
2015-04-23 19:36:30

a figure in 4d

a figure in 4d

goblashapa
2015-04-23 19:36:30

maybe a fourth dimension cube or something?

maybe a fourth dimension cube or something?

beanielove2
2015-04-23 19:36:30

4-dimensional cube!

4-dimensional cube!

Meimeijy
2015-04-23 19:36:30

hypercube

hypercube

beanielove2
2015-04-23 19:36:30

4-dimensional cube, or a tessarect.

4-dimensional cube, or a tessarect.

Wiggle Wam
2015-04-23 19:36:30

A fourth dimensional cube

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.

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?

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.

the dimension.

johngraham
2015-04-23 19:37:31

the dimentions

the dimentions

Meimeijy
2015-04-23 19:37:31

the dimensions

the dimensions

Silverfang
2015-04-23 19:37:31

Number of dimensions.

Number of dimensions.

Gwena
2015-04-23 19:37:31

dimension

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?

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...

0, 1, 2, 3...

Gwena
2015-04-23 19:38:20

0, 1, 2, 3

0, 1, 2, 3

osik
2015-04-23 19:38:20

goblashapa
2015-04-23 19:38:20

0,1,2,3 respectively

0,1,2,3 respectively

Wiggle Wam
2015-04-23 19:38:20

0,1,2,3

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.

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?

Now, what can we say about the next object in the sequence?

Gwena
2015-04-23 19:39:34

It has dimension 4

It has dimension 4

goblashapa
2015-04-23 19:39:34

It has four dimensions?

It has four dimensions?

Wiggle Wam
2015-04-23 19:39:34

4 dimensions

4 dimensions

kingda1
2015-04-23 19:39:34

Approaching the 4th dimension... *spooky music*

Approaching the 4th dimension... *spooky music*

gxah
2015-04-23 19:39:34

it has 4 dimensions

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:

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?

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

+ 1 dimension

goblashapa
2015-04-23 19:40:46

we ascend the dimensions? dunno.

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?

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

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?

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

We add another point

goblashapa
2015-04-23 19:42:18

Use another point

Use another point

kingda1
2015-04-23 19:42:18

add a point beside it

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.

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

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.

It's hard to do infinitely many things.

Wiggle Wam
2015-04-23 19:42:50

"extend" the point in some sense I guess

"extend" the point in some sense I guess

goblashapa
2015-04-23 19:43:21

make a string of points or something?

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?

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.

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.

We moosh the point one unit over, and keep the trail it leaves.

smbelcas
2015-04-23 19:43:46

smbelcas
2015-04-23 19:44:04

What do we do to go from a line segment to a square?

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

we moosh the line upwards

Gwena
2015-04-23 19:44:54

We smooth the line over for the same length as the line

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

'moosh' the line one unit over and keep the trail

TurtlePie
2015-04-23 19:44:54

Drag the line

Drag the line

osik
2015-04-23 19:44:54

smbelcas
2015-04-23 19:44:58

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?

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

Perpendicular to the line segment

TurtlePie
2015-04-23 19:46:01

Perpendicular direction

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.

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

(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?

Okay. What about going from a square to a cube?

champion999
2015-04-23 19:47:22

moosh it upwards!

moosh it upwards!

kingda1
2015-04-23 19:47:22

moosh again

moosh again

Gwena
2015-04-23 19:47:22

move a square perpendicularly for a specified distance

move a square perpendicularly for a specified distance

Wiggle Wam
2015-04-23 19:47:22

Moosh the square up

Moosh the square up

Philip7086
2015-04-23 19:47:22

moosh iy perpendicular to the plane of the square.

moosh iy perpendicular to the plane of the square.

goblashapa
2015-04-23 19:47:22

moosh the square over perpendicular to the square

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.

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

smbelcas
2015-04-23 19:47:35

And what comes next?

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?

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

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?

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.

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

We must move the cube perpendicularly in relation to itself

smbelcas
2015-04-23 19:49:16

Yes.

Yes.

smbelcas
2015-04-23 19:49:30

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

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:

Here are all the cubes we've made so far, for reference:

smbelcas
2015-04-23 19:50:03

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)?

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

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

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.

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

where is this going

smbelcas
2015-04-23 19:52:12

How do we make an $n$-cube?

How do we make an $n$-cube?

Wiggle Wam
2015-04-23 19:53:16

Moosh the n-1 cubes

Moosh the n-1 cubes

csmath
2015-04-23 19:53:16

take n-1 cube and move it perpendicularly into the nth dimension

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.

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

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

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

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.

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.

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.

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.

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

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...

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.

...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?

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.

Yes, it is.

smbelcas
2015-04-23 19:55:42

Historically, that was one of the most popular ways to do it.

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!

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?

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.

...No, but it takes a lot more practice to visualize.

smbelcas
2015-04-23 19:56:19

Anyway: 5-cube!

Anyway: 5-cube!

smbelcas
2015-04-23 19:56:22

smbelcas
2015-04-23 19:56:30

6-cube!

6-cube!

smbelcas
2015-04-23 19:56:34

hzbest
2015-04-23 19:56:50

Woah

Woah

Gwena
2015-04-23 19:56:50

That's beautiful.

That's beautiful.

csmath
2015-04-23 19:56:54

This is highly intimidating-looking.

This is highly intimidating-looking.

SirCalcsALot
2015-04-23 19:56:57

Really complicated!

Really complicated!

smbelcas
2015-04-23 19:57:04

Okay, that was just for fun. Sometimes I get a bit excited.

Okay, that was just for fun. Sometimes I get a bit excited.

azmath333
2015-04-23 19:57:08

7-cube?

7-cube?

smbelcas
2015-04-23 19:57:19

No, because at that point it gets too messy.

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.

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.

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?

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?

0 and 1?

somepersonoverhere
2015-04-23 19:58:57

(0) and (1)

(0) and (1)

SirCalcsALot
2015-04-23 19:58:57

$0$ and $1$

$0$ and $1$

Hydroxide
2015-04-23 19:58:57

0 and 1

0 and 1

smbelcas
2015-04-23 19:58:59

Yes, we put them at 0 and at 1.

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)?

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)

(0,0) (1,0) (0,1) (1,1)

thkim1011
2015-04-23 19:59:44

(0,0), (0,1), (1,0), (1,1)

(0,0), (0,1), (1,0), (1,1)

SirCalcsALot
2015-04-23 19:59:44

(0,0), (0,1), (1,1), (1,0)

(0,0), (0,1), (1,1), (1,0)

goblashapa
2015-04-23 19:59:44

(0,0), (1,0), (0,1), (1,1)

(0,0), (1,0), (0,1), (1,1)

somepersonoverhere
2015-04-23 19:59:44

(0, 0), (1, 0), (0, 1), (1, 1)

(0, 0), (1, 0), (0, 1), (1, 1)

champion999
2015-04-23 19:59:47

(0,0) (0,1) (1,0) (1,1)

(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).

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?

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.

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

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

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 ?

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 $2^n$

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.

A 3-cube has 8 corners. A 4-cube has 16 corners.

Darn
2015-04-23 20:03:02

2^n

2^n

champion999
2015-04-23 20:03:02

2^n

2^n

TurtlePie
2015-04-23 20:03:02

$2^n$

$2^n$

Wiggle Wam
2015-04-23 20:03:02

An n-cube has $2^n$ corners.

An n-cube has $2^n$ corners.

azmath333
2015-04-23 20:03:02

$2^n$

$2^n$

ImpossibleCube
2015-04-23 20:03:02

2^n corners for an n cube?

2^n corners for an n cube?

goblashapa
2015-04-23 20:03:02

number of corners: 2^n

number of corners: 2^n

Gwena
2015-04-23 20:03:02

The number of points in an n-cube is 2^n

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.

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 $(0,0,0),(0,0,1),(0,1,0),\ldots,(1,1,1)$.

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).

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

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

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

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

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

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

(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

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...

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?

...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.

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.

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

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.

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.

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

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.

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 $2^n$.

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?

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.

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

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?

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?

Does every $n$-tuple with 0/1 entries represent a corner of an $n$-cube?

TurtlePie
2015-04-23 20:11:17

YES

YES

smbelcas
2015-04-23 20:11:22

Explain why!

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.

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!

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!

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.

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:

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}$$

$$\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.

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.

I'll pass the answers through, column by column.

smbelcas
2015-04-23 20:15:21

For 2-cubes:

For 2-cubes:

Wiggle Wam
2015-04-23 20:15:26

4, 4,1

4, 4,1

mathwhiz918
2015-04-23 20:15:26

4 points in 2-cube

4 points in 2-cube

Gwena
2015-04-23 20:15:26

dim 2, points is 4

dim 2, points is 4

mathwhiz918
2015-04-23 20:15:30

4 lines in 2-cube

4 lines in 2-cube

mathwhiz918
2015-04-23 20:15:30

one square in 2-cube

one square in 2-cube

smbelcas
2015-04-23 20:15:44

What about 3-cubes or 4-cubes?

What about 3-cubes or 4-cubes?

goblashapa
2015-04-23 20:16:09

0 and 0

0 and 0

mathwhiz918
2015-04-23 20:16:48

0 3-cubes and 4-cubes for dim 0 and 1

0 3-cubes and 4-cubes for dim 0 and 1

smbelcas
2015-04-23 20:16:55

Also true.

Also true.

smbelcas
2015-04-23 20:17:00

Let's do the 3-cube column:

Let's do the 3-cube column:

Wiggle Wam
2015-04-23 20:17:11

8, 12, 6, 1

8, 12, 6, 1

goblashapa
2015-04-23 20:17:14

8 12 6 1 0

8 12 6 1 0

smbelcas
2015-04-23 20:17:55

What should the numbers be for the 4-cube?

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!)

(This is why I suggested at the beginning that scratch paper might be useful!)

Gwena
2015-04-23 20:19:17

16

16

goblashapa
2015-04-23 20:19:17

16 points 32 lines

16 points 32 lines

goblashapa
2015-04-23 20:20:16

i know its 1 4 cube

i know its 1 4 cube

smbelcas
2015-04-23 20:21:08

Anyone have any numbers for the 5-cube?

Anyone have any numbers for the 5-cube?

goblashapa
2015-04-23 20:21:43

32 points

32 points

Darn
2015-04-23 20:21:43

32, 80, 80, 40, 10, 1?

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?

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

um guessing

smbelcas
2015-04-23 20:22:17

I'm just not sure of all those numbers. Well, here is the updated table:

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...)

(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}$$

$$\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.

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.

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.)

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?

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)

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

Well lines is like $n\cdot 2^{n-1}$ i think

Gwena
2015-04-23 20:23:52

Well the ones will continue diagonally

Well the ones will continue diagonally

Gwena
2015-04-23 20:24:46

And so will the 0s

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)

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 $n$?

Any conjectures in general dimension $n$?

smbelcas
2015-04-23 20:27:12

Or about general face dimensions?

Or about general face dimensions?

Gwena
2015-04-23 20:27:18

Dimension n will have only one n-cube

Dimension n will have only one n-cube

Darn
2015-04-23 20:27:23

ooh squares looks like $\dbinom{n}{2}\cdot 2^{n-2}$

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?

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.

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.

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:

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

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

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'!

Okay! Now we're cookin'!

smbelcas
2015-04-23 20:30:50

Conjecture 1: $f_k(C_n) = 2f_k(C_{n-1})+f_{k-1}(C_{n-1})$.

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.)

(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.)

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?

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?

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

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

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.

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?

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

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

it gets duplicated and we record the trail

goblashapa
2015-04-23 20:36:23

moved one unit over and keep the trail

moved one unit over and keep the trail

Wiggle Wam
2015-04-23 20:36:23

It gets translated to form another corner point.

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.

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?

What happens to a line segment when we moosh?

Wiggle Wam
2015-04-23 20:37:40

forms a square

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

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.

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?

What happens to a square when we moosh?

goblashapa
2015-04-23 20:38:22

it becomes a 3 cube

it becomes a 3 cube

Gwena
2015-04-23 20:38:22

We get a 3-cube

We get a 3-cube

Wiggle Wam
2015-04-23 20:38:22

Forms a cube!!

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.

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

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?

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

it becomes a k+1 cube

Wiggle Wam
2015-04-23 20:39:43

Forms a k+1 cube.

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

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.

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?

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.

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).

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

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.

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

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.

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

$2f_kC_{n-1}$

$2f_kC_{n-1}$

smbelcas
2015-04-23 20:44:12

Yes!

Yes!

Gwena
2015-04-23 20:44:19

And each k-1 cube creates a new k cube

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.

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!

You've collectively proved Conjecture 1!

smbelcas
2015-04-23 20:45:09

Conjecture 2 is tougher.

Conjecture 2 is tougher.

smbelcas
2015-04-23 20:45:27

Any ideas?

Any ideas?

CInfinitesimal
2015-04-23 20:46:17

n choose k and the power of 2 makes me think of coordinates

n choose k and the power of 2 makes me think of coordinates

smbelcas
2015-04-23 20:47:49

Here we go:

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

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:)

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?

What do you all think?

smbelcas
2015-04-23 20:49:33

(I told you this one was tougher!)

(I told you this one was tougher!)

smbelcas
2015-04-23 20:50:06

Let me rephrase Darn's argument a little bit.

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 ${n\choose k}\dfrac{2^n}{2^k} = {n\choose k}2^{n-k}$.

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?

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.

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.

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!)

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.

{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.

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.

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!

Please ask questions!

goblashapa
2015-04-23 20:55:49

Can we apply if we are currently in 8th grade for this summer?

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!

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?

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).

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

where is this located

smbelcas
2015-04-23 20:57:20

This summer, MathILy is near Philadelphia and MathILy-Er is near Portland.

This summer, MathILy is near Philadelphia and MathILy-Er is near Portland.

Philip7086
2015-04-23 20:57:27

what does ILy stand for?

what does ILy stand for?

smbelcas
2015-04-23 20:57:33

Infused with Levity.

Infused with Levity.

CInfinitesimal
2015-04-23 20:57:51

How much calc does MathILy involve?

How much calc does MathILy involve?

smbelcas
2015-04-23 20:57:54

None!

None!

Wiggle Wam
2015-04-23 20:58:01

But the EAR would be due tomorrow! Could we still possibly get in?

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.

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.

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).

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

What is the EAR

smbelcas
2015-04-23 20:59:12

Exam Assessing Readiness!

Exam Assessing Readiness!

swamih
2015-04-23 20:59:24

Where can we find the EAR?

Where can we find the EAR?

smbelcas
2015-04-23 20:59:41

You can't. You have to ask for one personally.

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?

How much does the MathILy, MathILy-Er camp cost?

hzbest
2015-04-23 21:00:13

What's the tuition?

What's the tuition?

smbelcas
2015-04-23 21:00:27

$4600 for the five weeks, and there's need-based financial aid available.

$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?

How do you ask for EAR?

smbelcas
2015-04-23 21:00:46

Submit a Short Form on the website.

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?

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

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.

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.

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.

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.

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?"

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

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.

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.

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.

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?

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.

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.

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.

And 2-week long focused classes after that.

beanielove2
2015-04-23 21:06:24

What's Week of Chaos

What's Week of Chaos

smbelcas
2015-04-23 21:06:43

It's.... a week... full of chaos.

It's.... a week... full of chaos.

goblashapa
2015-04-23 21:06:45

Is lodging provided or where do we stay?

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.

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...

Chaos theory? Interested...

smbelcas
2015-04-23 21:08:00

Chaos theory is not

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?

Will you be one of the teachers?

smbelcas
2015-04-23 21:09:09

Yes.

Yes.

goblashapa
2015-04-23 21:09:28

At mathily or mathilyer

At mathily or mathilyer

smbelcas
2015-04-23 21:09:37

Hannah and I are both teaching at MathILy this summer.

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.

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.

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.

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?

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.

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.

I mean, not every day. But at least once per year.

smbelcas
2015-04-23 21:13:05

Or maybe once/week.

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?

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.

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.

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.

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.

And it's the Minion who sends you the EAR.

goblashapa
2015-04-23 21:14:10

Is it online?

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.

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.

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?

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.

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.)

(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

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).

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"

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?

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.

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.

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?

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.)

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!

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.

This room will be closing in a couple of minutes.