Mathematics - 2022-05-22 (page 2 of 2)

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Akiva Weinberger

9:17 AM

Here's a neat thingy

Find a set of tiles such that they can tile any arbitrarily large section of the plane, but such that they cannot tile the entire plane

(We can assume the tiles are polygons)

(I am being sneaky here though)

2 hours later…

one potato two potato

11:11 AM

Professor asked 'what is the relationship between connected, path connected, locally connected, locally path connected spaces?'. I answered '(locally) path connectedness implies (locally) connectedness and that's all. No other relationship exists.'

Professor then asked 'Ok then which space is locally path connected but not path connected?'. I hesitated because I was flustered and after a few seconds I answer 'Hmm.. something like comb space? It's path connected but not locally... oh the question is locally path connected but not path connected... so in locally path connected space, connectedness equivalent to path connectedness... so examples must be disconnected... Ah [0,1]\cup [2,3] is an example'

Now that I think about it, it's an easy question, but it was a difficult question back then

And I don't know why I'm talking this now in here. Just ignore it as usual

1 hour later…

12:21 PM

in Constructive Feedback, 4 mins ago, by Shaun

Please would someone shed light on why the following was downvoted?

-1

Q: In what sense is $\Bbb R(x)$ an "instantiation" of the hyperreals?

I'm teaching myself about hyperreal numbers. My main motivation for doing so is that they include infinite numbers, whose existence I hear disputed & doubted often as "quantifying the unquantifiable". In this YouTube video, around twelve minutes in, it states that $$\Bbb R(x)=\left\{ \frac{f(x)}...

polynomials terminology real-numbers nonstandard-analysis nonarchimedian-analysis

1:19 PM

@AkivaWeinberger Just use hexagonal donuts of sizes { 2^−k : k∈ℕ }.

I don't see anything sneaky. This is trivial once you know that compactness more or less excludes all possible tile sets that have a positive minimum size.

2:12 PM

@AkivaWeinberger: Wait, if you can just use infinitely many different sizes, then my answer fails.

Akiva Weinberger

2:28 PM

@user21820 You sure you want no minimum?

@AkivaWeinberger I'll think about it again later. Busy with something else now. =)

@AkivaWeinberger Here's some acapella from Berklee for you. Stevie Wonder's I Wish

3:08 PM

@AkivaWeinberger You're right; having a minimum size makes it work. You just need a sequence S of congruent shapes of exponentially increasing size where each S[k] is the largest piece that can fill a notch in S[k+1]. Then there is no way to fill the notch in the smallest S[m] in any configuration, but you can just use a single sufficiently big member of S to cover any given finite region.

Akiva Weinberger

@user21820 That works. The example I had thought of is regular pentagons of all the integer sidelengths.

@AkivaWeinberger Lol that was easier than I thought...

Now, what if you want to have a configuration with total area r for every positive rational r?

Is it still possible to be unable to tile the whole plane?

Akiva Weinberger

Does the Apollonian gasket count as a tiling?

user image

@AkivaWeinberger Yes, But it can tile the whole plane.

Akiva Weinberger

Sure. I think if you have arbitrarily small tiles you can repeat the same procedure as the Apollonian gasket and it'll work

3:14 PM

The only way to avoid that would be to have pieces with arbitarily small area but not necessarily small diameter.

Akiva Weinberger

Oh… hm

Good point

So, no, I don't think it's obvious.

Akiva Weinberger

And "can tile an arbitrarily large area" means, for any radius $r$, we can cover a circle of radius $r$, right?

This is tricky

I kinda want to be able to take a "limit" of a sequence of tilings of a finite area

or, rather, say any set of tilings of the finite area has a convergent subsequence

@AkivaWeinberger Yes, but I guess we need extra conditions to make it interesting. Otherwise we could just have curved strips of area 1/2^k but diameter 3^k/2^k, plus pentagons with natural sides.

Akiva Weinberger

Oh, that's a good example. That was simpler than I thought

I guess I was thinking "maximum area but no minimum area"

(and also minimum diameter, or else we can do an Apollo)

3:24 PM

I think there should be a much more tricky version of your question, but I can't think of a good one right now.

Akiva Weinberger

You should probably change your username, by the way

(there should be a button here: stackexchange.com/users/1131127/user21820)

otherwise it's hard to remember who you are

@AkivaWeinberger Oh, why? I'm kind of the only user that you would need to remember, because I'm the 'only one' who stuck to the default username. =P

Prithu biswas leftmse

@AkivaWeinberger the number "21280" is forever engraved in my brain.

@AkivaWeinberger And I did know about that feature, but thanks!

@Prithubiswasleftmse Ironically, you mistyped it, but the multiset of digits is correct. XD

surprisingly, I remember 21820 too :)

Prithu biswas leftmse

3:31 PM

@user21820 lol.

@AkivaWeinberger: How about this:

> Question: Consider a set S of tiles. Is it possible that S cannot tile the whole plane but for any finite Jordan-measurable region R there is some Jordan-measurable region R'⊇R such that Area(R') ≤ Area(R)+1 and R' can be tiled by S?

Akiva Weinberger

Does that kill the pentagons example?

You can try...

It should.

=P

Akiva Weinberger

Interesting

I'm thinking, to define "tile", that these are disjoint open sets where the union of their closures is the region

@AkivaWeinberger Yes that's fine.

@leslietownes: You may want to join us in finding weird tricky tiling questions.

Akiva Weinberger

3:55 PM

Wait I was gonna say something else but I think I have an idea

Unit-size squares with notches

One set of noches for every polygon made of squares that you can want

or actually we only need it to make nxn squares

I don't really understand your idea; how are you going to approximately cover any given region with only unit area discrepancy?

Akiva Weinberger

Each tile has notches specifying the size of the large square the tile will be used in, and where the tile fits inside it

and the "edge" tiles have some notch that's incompatible with everything else somehow

The given region need not be a square of integer size.

Are you going to encode every polygon?

Akiva Weinberger

Sure, but a subset of that square will work

Would it? I had the "Area(R') ≤ Area(R)+1" condition.

Akiva Weinberger

3:58 PM

Oh, hm

Wait

Does that mean that even just a square tile does not satisfy your criterion

Yes.

=D

Akiva Weinberger

Erg

Now you get the hard point.

a single (connected) shape that nonperiodically tiles the plane

2 hours later…

6:15 PM

Lemma: "Let a be a non zero integer and b an integer. If a|b, then gcd(a,b)=|a|.". Is the following proof correct? Proof: "By definition of gcd, gcd(a,b)|a and gcd(a,b)|b. Since a|a and by hypothesis a|b, by definition of gcd it is a|gcd(a,b). A previous result assures that x|y and y|x implies x=y or x=-y; so, from gcd(a,b)|a and a|gcd(a,b), it is gcd(a,b)=a or gcd(a,b)=-a, that is gcd(a,b)=|a|."

6:45 PM

that looks ok to me, if the 'by definition of gcd' things are known from your definition of gcd. [i'm not sure, for example, if some definitions of gcd make it immediately clear that a common divisor of a and b must divide the gcd, as opposed to, for example, simply be no greater than it]

huh, only four people in the chat? is this the chat.se version of a netsplit?

thanks leslie, the definition I have is the following: "x is called greatest common divisor of y and z if x|y and x|z and for any t such that t|y and t|z it is t|x.". The numbers x,y,z and are all integers

ok, it certainly follows from that definition.

is gcd defined up to sign, or is it defined to be positive?

note that whether gcd(a,b) = a or gcd(a,b) = -a might depend on the sign of a and that aspect of the definition of gcd. they won't both be true.

You are right, x must be >=1 and y and z must be not both zero

Akiva Weinberger

7:08 PM

@geocalc33 OK, Einstein

*ein stein

zwei steine

speaking of which, love a beer right now

7:45 PM

is it summery up there? i could go for a white beer right about now.

beautiful outside

i'm going to the el cerrito pool for a swim in a while

(of course, the water is like bath water there, so that's not saying much)

8:17 PM

another downvote

@MadSpaces If my memory hasn't failed me, a few days ago you posted a question about using forms to prove $\text{rot}(\text{rot}\, \vec F) = \nabla(\text{div}\,\vec F) - \Delta\vec F$. Personally, I have never had much use for such identities, but, yes, you can do it rather cleanly with forms. You represent $\vec F$ as a $1$-form and then you do lots of stuff with $\star$ and $d$.

In particular, you use $\Delta = d\delta + \delta d$ ($\delta$ being the adjoint of $d$), together with appropriate formulas like $\delta = \pm\star d\star$.

It works out in a few lines.

If you're going to do it "classically," my recommendation would be to just check it (again in a few lines) for $\vec F = f\hat i$ and then appeal to linearity. No crazy vector identities, which are particularly suspicious when you say you can use $\nabla$ in those formulas.

(To elaborate just a little bit, not worrying about signs, note that if $\omega$ is the $1$-form corresponding to $\vec F$, then $\text{rot}\,\vec F$ corresponds to $\star d\omega$ and so $\text{rot}(\text{rot}\,\vec F)$ corresponds to $\star d\star d\omega$. $\text{div}\,\vec F$ corresponds to $\star d\star\omega$. I leave it to you do check all this carefully and proceed.

@copper.hat I am so embarrassed. I downvoted an answer from some years ago completely by accident. I had no idea I'd done it, but then it wouldn't let me undo it a few hours later. Sigh.

@TedShifrin That's funny! No bother, I don't care about the rep. Just if there is something wrong (which is quite likely) I would rather effect a repair.

Yeah, the 5-10 downvotes on my stuff have taken a holiday for now. I don't think they were done for a substantive reason, although one of them may have been the OP who ultimately decided what I said wasn't helpful after all.

I just want my jump suit :-)

8:34 PM

I think you're spreading rumors. I have no evidence that I'm ever going to get one.

Does MSE even necessarily have a correct address for us?

I have no reason to believe they have anything other than a city.

if they have my city, it's about a decade out of date.

you will get an email from MSE's concierge for elite patron services for updated info.

Well, I updated my city when I moved. That doesn't seem to be asking too much of us to accomplish.

"Concierge for elite patron services"? Where did you concoct that nonsense?

well, if i do that then i can't fraudulently vote in numerous elections.

i never update any information

same place i get all my other nonsense.

Well, your vote would make a bigger difference in Iowa than in either Massachusetts or California.

today, i'm having hummus with triscuits as g-d intended.

i did like voting in iowa.

and in michigan.

8:41 PM

I wonder if MI is going to end up as bonkers crazy as PA. My political science friend who now lives there insists not.

it would surprise me. aspects of the labor movement were basically born there.

in the really rural counties you see the 'southernification' of political rhetoric but it's not many people who live in those.

Well, don't forget they tried to kill the governor.

it doesn't surprise me that MI harbors really unhinged people. i just hope it's not a critical mass of them.

Heyo Ted, Leslie

Hi ho, Sha.

8:43 PM

some of their republican congressmen are middle of the road types who would not have been out of place on either side of a 1990s caucus.

oh no, here comes geometry. :) hi sha

No, it's all algebra. Not geometry.

Hmmmm

I spent the entire day writing out all the symmetries of the cube

because that's an exercise for my students tomorrow

is that a cover story for a gambling addiction?

i spent all day at the casino, researching the cube.

hahahahaha xD

i need to do it for my students.

8:45 PM

hahahah xD

in any case, I appreciate the geometry aspect of discrete groups now much more

Oh, I always enjoyed teaching the concrete symmetries in my algebra course. I made the kids make their own tetrahedra, but I had all the other shapes for them to use/borrow.

then some spherical trigonometry at the pool hall.

I'd like to see you use the spherical law of sines/cosines at the pool table.

my hs physics teacher said "knowledge of pool is a sign of a misspent youth."

he was a fossil.

@TedShifrin for me it's the most fun part of the course

though when I took it as a student I was much more focused on the formalism of groups

8:47 PM

Group actions rule.

hell yea

I made up some good exercises when I wrote my book and/or taught the course.

9:04 PM

the internet makes critical masses of like minded morons

rhetoric matters more than reason

Akiva Weinberger

How's it going

user image

I recently learned that if you have a square ABCD with corners A and C opposite and B and D opposite, you can find two disjoint connected subsets of the square, the first containing A and C and the second containing B and D

which is totally intuition-breaking

but the key is that the subsets need not be closed

I suspect (but have not yet proven) that if they're closed you can't do this

(some sort of homology argument maybe?)

Wait, I have proven that, ages ago.

Yeah OK so you need nonclosed thingies to make that work. In the picture, each set (black and purple) only contains one point of the left edge (the top and bottom points respectively)

are those curves continuous on $(0,1]$?

Akiva Weinberger

Should be

They were basically messed up versions of $\sin(1/x)$

it is cute, but disturbing.

I agree that the statement chafes at one's intuition.

Akiva Weinberger

9:14 PM

done so that the "teeth" mesh perfectly and never intersect

It demonstrates how weird the notion of connectedness is (if you split it into two pieces, one piece contains a limit point of the other)

^that's equivalent to the usual definition but I think I like it better for understanding why we define it like that?

Well, it just shows that set that are neither open nor closed can be very f***ed up.

Akiva Weinberger

Like me!

DogAteMy, here is a whacko problem of a more elementary nature. Do you see any elegant solution? Mine uses trig (law of cosines and sines) and numerical solution of a transcendental equation.

Actually, a second degree Taylor polynomial is more than sufficiently accurate.

@TedShifrin (I hope u don't mind the ping) but are you aware of gyrovectors?

Nope.

9:24 PM

@TedShifrin Sigh ... when u have great idea but the mathematics used is too obscure

I assume "u" refers to you and not to me.

Akiva Weinberger

I’m not sure I understand the question. A triangle equals a sector?

Areas.

Akiva Weinberger

Ah.

Is c halfway up?

@TedShifrin yes

9:26 PM

But my suggested modification (adding the same thing to both) makes it more tractable.

No, you have to read his text, but he gives you the radius is 30 and $ab$ is $0.5$.

Akiva Weinberger

Oh, I see

@TedShifrin Good call

I wonder if there's a more elegant approach than mine.

Akiva Weinberger

Yeah from there it's algebra I guess

You're usually far clevererererer at such things.

No, trigonometry + transcendental equation.

Akiva Weinberger

Sure, but just manipulation to get to that point

9:28 PM

Oh, sure.

Akiva Weinberger

Yeah sometimes you just gotta do the braindead thing and not worry about being clever

I will ask the OP from whence this problem comes.

Akiva Weinberger

Do you know the pizza theorem?

Um, hum a few bars.

Akiva Weinberger

Cut a circle into $2n$ pieces with straight lines, but with the lines all passing through some point $p$ instead of the center, and making equal angles at $p$

Give me every other slice and give you the remaining slices

9:32 PM

Oh, and we get the same amount of pizza.

Akiva Weinberger

If $n$ is an even number that's greater than or equal to 4, we always get the same area

No, I've never encountered this.

oh this is the weed theorem

Akiva Weinberger

otherwise, no unless one cut passes through the center

This has the feeling of some classical Möbius stuff.

I wonder if it's in Pedoe's book on circles.

Akiva Weinberger

9:34 PM

Möbius? I only know his band

Morbius? it's only the best movie of all time.

Linear fractional transformations.

Akiva Weinberger

Oh, I see

Those mess with areas and lines though

Well, yes.

nah, they send lines to lines. you just need to change your conception of lines, man.

Akiva Weinberger

9:36 PM

I basically wrote the circle translated horizontally by $\alpha$ in polar coordinates and saw that $r^2$ was a quadratic in $\cos$ plus an odd function of $\cos$

Oh, you have to still make all the cuts at equal angles?

my daughter was put to her nap about 20 minutes ago and is still babbling and singing to herself in a darkened room. at what point does this become my problem?

Akiva Weinberger

and from there you get all the things in the integral canceling

21 minutes ago.

Akiva Weinberger

@TedShifrin Yeah

They don't cut the circumference into equal pieces though

9:37 PM

Right.

i thought she was talking to the cat, which would have made it sort of normal (for her), but then the cat walked into my office while she was doing this

The cat walked into your office and said, "You have to do something about that crazy girl."

that's basically the look that i got.

the line i use with my wife is "it's your DNA," meaning it's not my fault. except, my daughter picked this up from me. so sometimes she'll misbehave, and then point at my wife and say "it's your DNA."

Akiva Weinberger

Is it not also yours?

well, yes, but i'm blaming her half.

9:40 PM

And we in this chatroom know whose DNA is more to blame.

Akiva Weinberger

Doctor says, "All you DNA is backwards"

"And?"

i'll allow that. that's funny.

she was going after the cat with a pair of tweezers earlier, pulling out tiny tufts. the cat let her do this for a surprisingly long time, then cuffed her in the face.

Akiva Weinberger

I think cats don't fully understand humans until they encounter a baby

and then everything starts to fall into place

that's why this dumb ape can't groom himself properly.

Akiva Weinberger

I think saying that made me sound like a cat trying to understand humans

9:44 PM

on the internet, nobody knows if you're a cat

elon knows

The h*** with that b**ad.

Hmm, I can't use asterisks very effectively cuz bold and italic.

ted and i agree again.

Akiva Weinberger

Guess you'll just have to swear

i was tempted to respond to his twitter request for legal counsel except i was afraid someone might end up actually reading it and getting me in trouble.

Akiva Weinberger

9:48 PM

Is there a formula for $\binom n1+\dotsb+\binom nk$? No, right?

partial sums of rows of pascal's triangle?

There is if both top and bottom increase by $1$ each time.

arbitrary partial sums?

Akiva Weinberger

Yeah

I know not of any formula, but there are lots of obscure things out there I know not.

9:50 PM

i don't know of one. this feeds slightly into my hazy plans at a bijective inquiry into that weird formula for congruences mod p (= 17) that you raised a while ago.

Akiva Weinberger

I'm actually willing to conjecture that no formula exists, and go on a quest in search of tools that let me prove such things

How could one possibly prove non-existence of a formula?

ask doron zeilberger. if he doesn't know, his answer may be colorful.

You just define a new function by that sum and presto you have a formula.

Like the Lambert W function, for example.

Akiva Weinberger

I mean, in terms of the usual pieces

9:52 PM

So you want an analog of Liouville's theorem for non-integrability in elementary terms?

you can certainly prove nonexistence with a whole lot more hypotheses on what existence would be.

whoville?

Akiva Weinberger

@TedShifrin I don't think I know that one

Maybe there's a way to show that if this is elementary, then so is erf(x)

and I don't know how we know that erf is nonelementary

whoville proved it.

sharp guy. some of the people in the 19th century weren't dunces.

It's called differential algebra ... the entrance of field/Galois theory into questions of non-elementary antiderivatives.

Akiva Weinberger

I’m basically calculating percentiles, with my formula

'cause binomials are approximately normally distributed

9:55 PM

I'm just saying he developed an algebraic theory to encapsulate the notion of integrability in elementary terms. You need to encapsulate your notions.

you can't just have your notions going about willy-nilly, and flapping in the breeze

Akiva Weinberger

I must have non-nil will?

willy nilly is short for William Nilliam

or Wilhelmina Nilliam

Bill Nilly, really.

10:27 PM

By the way, @Akiva, that OP was trying to do some classical Ptolemaic astronomy ... so no reason to expect anything different from the trig, etc., I was getting.

Akiva Weinberger

Oh, interesting

Geometry (and classical geometry specifically) was never a strong suit of mine

maimonedes comes up (as he tends to)

Akiva Weinberger

which is odd given how much I like pictures

Well, this really isn't much geometry that I can see ... definitely trigonometry (not surprising if we're doing astronomy).

Akiva Weinberger

@bumblebee brought up something called Alhazen's problem, that I hadn't heard of before

of bouncing a projectile off a circle to land it at another spot

Alhazen's problem

Alhazen's problem, also known as Alhazen's billiard problem, is a mathematical problem in geometrical optics first formulated by Ptolemy in 150 AD. It is named for the 11th-century Arab mathematician Alhazen (Ibn al-Haytham) who presented a geometric solution in his Book of Optics. The algebraic solution involves quartic equations and was found in 1965 by Jack M. Elkin. == Geometric formulation == The problem comprises drawing lines from two points, meeting at a third point on the circumference of a circle and making equal angles with the normal at that point (specular reflection). Thus, its main...

10:36 PM

oh, i've heard of that, in some math history book.

Akiva Weinberger

Wikipedia tells me it ends up being a quartic, which doesn't really sound all that fun

yeah, i think that's how it came up in the book, as an example of a situation that leads to a quartic.

alhazen was very prolific.

Akiva Weinberger

A lot of math like that is more focused on calculating quantities than proving theorems.

I guess they call that "applied math"

but also they used to just call that "math"

and it doesn't appeal to me as much

A few of my quite difficult papers calculated some interesting geometric quantities (like how many asymptotic flex points are on a smooth projective surface of degree $d$) ...

It's plenty full of theory.

a friend of mine who grew up in mainland china is always complaining about how ancient chinese mathematics (which he reveres, and rightly so) was too preoccupied with "practical things" and not theoretical enough.

10:38 PM

So don't be so derogatory about actually computing an integer.

The computation of characteristic classes of vector bundles is anything but mundane and useless.

Akiva Weinberger

Well, integer. Here it's a point on a circle

> Alhazen was able to show, by purely geometrical arguments, that the location of the point on the circular mirror at which the reflection must occur lies at the intersection of that circle with a hyperbola.

Oh that sounds interesting though^

sharp guy. they weren't all dunces in the 11th century.

Akiva Weinberger

I wonder what hyperbola

Precisely. Proving a locus you define one way can be computed in an entirely different way is part of what we did in our paper, too.

Akiva Weinberger

Were they all dunces in the 21st century?

Or is that yet to be determined

10:40 PM

at that time my ancestors were probably solving the problem of how to get enough gruel to avoid dying of malnutrition.

akiva: yes.

The world is full of dunces, now.

Well, my work was in the 20th, so who knows.

well, dunces can find each other easily now

a lot of that classical problem solving is really, really sharp. even if the algebra gets bad, they'll say, oh, with this curve you can find it. or whatever. like inventing the W function.

there were maybe 5 non-dunces at any given time.

throws potato at copper.hat

Akiva Weinberger

Potoooooooo or variations of Pot-8-Os (1773 – November 1800) was an 18th-century thoroughbred racehorse who won over 30 races and defeated some of the greatest racehorses of the time. He went on to be a sire. He is best known for the unusual spelling of his name, pronounced 'Potatoes'. == Background == Potoooooooo (also spelled Pot-8-Os, Pot8Os, Pot8O's or Pot 8 Os from various sources) was a chestnut colt bred by Willoughby Bertie, 4th Earl of Abingdon, in 1773. He was sired by the undefeated Eclipse. He was the first foal out of Sportsmistress, who was sired by Warren's Sportsman and traced to...

Pot "eight 'o's"

Potoooooooo

Racehorse

a pun is always welcome, but that's taking it to an extreme. thumbs down from me.

Akiva Weinberger

10:43 PM

Horse doesn't have thumbs

This person/post has made me literally scream. Help.

we do have a 1700s painting of a horse on our refrigerator.

or a paper-printed reproduction of one, rather.

ted, your comment is what i would have said. it's not nonsense to think in this direction but they do need to slow down.

Akiva Weinberger

@TedShifrin I think a better way to write "this makes absolutely no sense" is "you made a type error"

for this reason, i won't talk about operator valued cosines.

Yes, I know that we can put matrices into a power series and define analytic functions. I honestly don't think this person is capable of that.

10:46 PM

matrices implies the existence of a matrice.

Well, and let's just throw determinants in because that's what I'm used to with that notation.

Akiva Weinberger

@TedShifrin But, like, the question is just "I can define this quantity loosely inspired by a notational game, does it have an interpretation"

i also very much agree with not using |\cdot| for determinant.

it's the sky falling in on our heads.

I never did when teaching. Use norm and absolute value interchangeably leads to enough disasters.

Akiva Weinberger

If they understand it's a notational game, and I think they do, then I don't think there's an issue here

They never made a statement of fact

10:49 PM

Oh, they realize that the cosine will be a matrix with that definition. I guess I'm going to remove my yelling. It's just pointless.

separate from series or anything else you'd want this thing to be associated with something like a partial isometry connecting A with B, maybe followed by something. the formula won't work, even if you fiddle with functional calculus

as i used to tell my officemate, we actually have whole classes in this stuff for a reason.

Well, I removed my ranting. The person clearly knew it was a matrix-valued inner product. The right question is: With that definition, what algebraic properties can you derive, if any?

And you certainly have to remove all singular matrices from the domain.

that's where the partial part of the partial isometry would come from. but it does f up the formula.

and, not a rabbit hole worth diving down without some goal in mind beyond, what does [fiddlesticks] mean.

| | should definitely not be the determinant. it might be spectral radius. there's no multiplicative piece there.

fiddlesticks.

washingtonpost.com/science/2022/05/21/… is a cute story.

i love how the prof did that so often he can't personally remember the subject of the story.

go bears

Very cool. And she wasn't scared of roaches or spiders. All power to her.

11:06 PM

my daughter has a very realistic plastic cockroach that she places around the house. i have no idea where she got it.

hopefully she doesn't go down this path.

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