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/sci/ - Science & Math

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5545996 No.5545996 [Reply] [Original]

There is a proper definition for a circle that is found in most calculus books.

Without googling or looking it up in your book, define a circle without using the equation/formula of a circle or any facts about any other geometrical shapes ("points" are allowed.)

Your definition must exactly define a circle and must not be applicable to any other shape.

>> No.5545998

1) take a point
2) a circle is all points of equal distance from that point

This is something most people learn in 4th grade

>> No.5546000

the set of points with the same distance to a given point

x^2+y^2 = r^2

sin(t), cos(t)

>> No.5546010

>>5545998
This.
What kind of challenge is this? I know /sci/ is pretty stupid but not this stupid.
The equation simply translates this fact into coordinates using the Pythagorean theorem.

>> No.5546012

>>5545998
You have to mention that you're in R^2 or it's ambiguous.
That also defines a sphere, two points on a line, every hypersphere in dimensions 4 onward, etc...

>> No.5546035

>>5545998
this guy
>>5546012

>>5546000
you broke the "without using the equation/formula of a circle" restriction.

>> No.5546049

Another definition:
a circle is a maximal connected curve with constant curvature

>> No.5546062

>>5546049
This is the one I like the most (OP.)
There's an rigorous/proper way to state it, but it means exactly what you're saying.

>> No.5546066
File: 20 KB, 391x411, Weather the storm QT.jpg [View same] [iqdb] [saucenao] [google]
5546066

>>5546062
>There's an rigorous/proper way to state it, but it means exactly what you're saying.

It is means exactly what he says, then it is rigorous, unless what he said isn't exactly the same.

Step up yo game nigga

>> No.5546068

>>5545996
>implying there is only one equation that describes a circle

>> No.5546074

>>5546066
What I meant is that the definition I read also includes the definitions of maximal and curvature, and defines it all using proper notation.
Maybe I should have said "in-depth" instead of "rigorous?"

>> No.5546077

>>5546068
>inferring something I didn't imply
I didn't.
I actually wanted to see how many different definitions /sci/ could come up with.

>> No.5546081

>>5546074
>proper notation

Trust me, once you start reading through papers you will learn soon enough there is no 'proper notation' for anything, i've even seen shit like "let t be the ratio of a circles circumference to it's diameter" because pi was already used as some mapping

>> No.5546082

>>5546077

Yes you did.

>the equation/formula of a circle
>singular

>> No.5546084

>>5546081
You mean I'm learning all this notation and people don't even use it much?
fuuuuck.

>> No.5546088

>>5546082
Oh, my bad. I misread your green text as "one definition" instead of "one equation."
I was specifically thinking of x^2 + y^2 = r^2 in the OP.

>> No.5546092

>>5546084
You'll find many notations for almost anything

>> No.5546098

A circle is the limit of an n-gon of n equal sides meeting at equal angles as n -> infinity

>> No.5546109

>>5546098
Oooh, that's a cool one.
Kinda sly when it comes to the "no facts about other shapes" restriction, but I really like it.

>> No.5546119

<span class="math">
\phi \in [0;2\pi] \colon graph \{ e^{\i \phi}\}[/spoiler]

>> No.5546121

<span class="math">
\phi \in [0;2\pi] \colon graph \{ e^{i \phi}\}[/spoiler]

>> No.5546133

>>5546098
>realize that curves are a bunch of really small lines after noticing a similar phenomenon in my train set when I was 7
>"okay class, how many sides are in a circle"
>"infinity"
>entire class snickers at me
If only they knew how wrong they were.

>> No.5546141

>>5546133
you're wrong, moron, infinity isn't a number (inb4 some smart ass starts talking about extended complex plane or some other bullshit completely irrelevant)

>> No.5546147

>>5546141
> completely irrelevant because it shows I'm talking out of my ass
do you even wipe your own ass

>> No.5546149

<span class="math">\mathbb{R} / \mathbb{Z}[/spoiler]

>> No.5546150

>>5546141
Do you really expect a 2nd grader to understand the concept of a limit?

>> No.5546153

>>5546141
You're embarassing yourself.

>> No.5546158

>>5546150
I expect a 2nd grader not to try to be 'that guy' who keeps acting like he knows shit when he doesn't

>> No.5546165

How many five-letter words(with repetitions allowed) can be made from 26 letters of the alphabet if no two adjacent letters may be the same?

I got 26*(25)^4. Anyone else have any other answers?

My logic: For the first position, there are 26 choices to choose from. From the second, there are now 25(since you can't have the same letter there). Now, for the third. Assume you start again with 26 letters. You can use the letter from the first position, BUT you can't use the letter from the second. Hence, there are 25 letters to choose from for the third. This pattern continues for the fourth and fifth positions of the word.

Is this a good solution?

>> No.5546164

>>5546158
Hey guys, we have someone really cool overhear. He's better than 2nd graders he says.

>> No.5546170

lol wtf i thought this at uni yesterday, are you scanning my brain??

okey so taken any point in two dimensional surface, say c_0, then array of points {p_1..p_inf} which satisfy d(c_0,p)=r is said to be a circle with radius r.

>> No.5546179

>>5546165
Oooh, that's an interesting one.

(Yeah, OP here, if ya'll want to just keep making new challenges like this guy, go for it. I think it's fun.)

Are you saying no two adjacent letters in the word are the same, or "if a letter is picked for this combination, you can't choose any letters that were adjacent to it in the alphabet?"

>> No.5546183

>>5546165
Isn't that just 26 choose 5?

>> No.5546186

>>5546170
Is c_0 the center/origin of the circle?
If so, that sounds like it works.

>> No.5546184

It is the set of all orbits on the interval (0,1]

>> No.5546192

>>5546183
Nay, not if I understand him right. That'd allow for same-type adjacency.
For example, "MOON," which is illegal.
You need to remove all possible combinations with same-type adjacencies from the 26 choose 5.

>> No.5546193

a circle is a regular agon...

>> No.5546196

>>5546192
So would ABCA be allowed, but AABC would not?

>> No.5546197

>>5546183
No. You're choosing five different letters there. This actually ALLOWS repeats. For example, I can have:

abaca. I can also have acqkq. That's perfectly fine. I just can't have aabca because those two letters in the first two units disobeys the law.

>>5546179
If you pick any letter, the same letter can't appear next to it. So, aabcd is incorrect. abaca is fine, though.

>> No.5546198

>>5546196
Five letters, but yes. ABCAB would be allowed, but not AABCA.

I have a shitload of problems of this stuff, guys.

>> No.5546199

n+k-1 choose k is the choice function when repetion is allowed.
So how would you figure it such that 2 elements of the set are not allowed in sequence

>> No.5546202

>>5546192
Doesn't 26 choose 5 imply that you pick five letters from the 26? Since no letter repeats(by how the English language is designed), you'll have five distinct letters always.

>> No.5546204
File: 224 KB, 1600x1200, DSC_0001.jpg [View same] [iqdb] [saucenao] [google]
5546204

>>5546170 >are you scanning my brain??

Yes.

>> No.5546206

>>5546204
Timestamp that shit and I'll shit bricks.

>> No.5546208

>>5546165
Yes, it's right.

>> No.5546215

>>5546208
Cool. Someone else can dispute my solution or put one of their own. I don't mind either way.

Here's another combinatorics problem:

How many even five-digit numbers are there, provided that there are no leading zeroes(i.e., 00174).

I solved this one and others, so I have the answer.

>> No.5546218

>>5546215
9*10*10*10*5

>> No.5546221

>>5546165
So, thinking out loud here. Let's go through this one slot/step at a time.

A) For the first slot, you can choose any letter.
B) For the intermediate slots, you can choose all but the one before and all but the one that follows.
C) And for the last slot, you can choose all but the one to the left.

So A always has 26 choices
and C always has 25.
B could have 25 or 24 depending on whether the ones on it's sides are the same or not...
Very interesting...

So at most you have 26,25,25,25,25
and at least you have 26,24,24,24,25
Your 26*25^4 seems to show that pretty well, but how do we know if we're excluding rearranges that would result in adjacencies?
For example,
ababa is correct, but if rearranged, aaabb is illegal.
Does 26*25^4 exclude those possibilities?

>> No.5546224

>>5546218
Why the last one times 5 ?

>> No.5546226

>>5546224
EVEN

>> No.5546228

>>5546192
Combinations are without repitition. 26 choose 5 describes the number of ways to pick 5 different letters from the alphabet if the order doesn't matter. Your first solution seems right to me.

>> No.5546236

>>5546199
That formula is no good to begin with since the question is how many words can be made, implying the order of the letters matters. Permutations is the way to go.

>> No.5546243

>>5546218
Good.
>>5546224
In the last slot, there can only be five numbers: 0, 2, 4, 6, 8. In the first slot, only 9 -- we exclude 0.
>>5546221
Interesting way to think about it.

In truth, it's more so 'how' you think about it, I guess. In each slot, I'm deliberately taking a letter out, putting one back in, taking one out, etc. Can you use that same logic with the other/secondary combination problem?

For example, in the 9*10^3*5, if you have 12306(a possible arrangement), could you 'rearrange' that to 01236? I don't think so.

>> No.5546245

>>5546243
Here's another. Keeping it within a decent range.

Let there be ten basketball players. How many five-man teams can be created, given that the weakest player AND the strongest player must be on the team? How many ways are there to pick a five-man team from the ten basketball players, in general?

>> No.5546246

>>5546221
First slot any letter, next slot, any letter but the one before, repeat.

You're overcomplicating things by considering the next letter before it even has been picked.

>> No.5546252

>>5546246
>>5546221
Oh and the illegal rearranges don't matter since we aren't counting those and nothing gets rearranged.

>> No.5546253

>>5546221
>So A always has 26 choices
>and C always has 25.
>B could have 25 or 24 depending on whether the ones on it's sides are the same or not...
>Very interesting...

I want to note this, by the way. Forgot to in my other post.

For the first slot, if you choose any letter, there are 26 choices. Cool. In the second slot, there are 25 choices. Now, in the third slot, you take out whatever letter you used to fill in the second slot; you can also put 'back in' the letter from the first slot, resulting in 25(again). For the fourth, the pattern repeats. Why do you say 24 slots? Not disrespecting/or antagonizing you, just want to see where you're coming from.

>> No.5546251

>>5546245
Where are you getting these questions?
This is like the first page of a combinatorics book.
Anyway
1.(8 3)
2. (10 5)
(a b) being a choose b.

>> No.5546260

>>5546251
Aha. From my combinatorics and graph theory book. I'm not giving the harder ones or anything.

How many five-digit numbers are there that are the same when the order of the digits is inverted? For example, 15251

>> No.5546264

>>5546253
I let myself overcomplicate and go out of order. It was silly.
I was thinking of a middle slot where the slot to the left and right had already been chosen.
So if you're looking the second slot in [c,_,t] you only have 24 choices since you can't choose c or t.
If you have [c,_,c] you have 25 choices since only c is restricted.
It was a silly mistake where I broke my own rule.

>> No.5546267

>>5546251
I think you need to divide by 2! since any way to pick one group already creates the other group.

>> No.5546270

>>5546264
Ah, okay.

Don't call it a silly mistake. I've learned(painfully) that combinatorics is a pretty unforgiving and brutal course, mentally. It stretches your mind, that's for sure. Multiple outcomes have to be decided, judged, thought of, etc.

>> No.5546275

>>5546267
Nope. It's correct.

For the weakest player, it's 1 choose 1. For the strongest, it's 1 choose 1. Then, there are eight players left. You can choose any three of them since order doesn't matter; hence, 8 choose 3.

For the second one, 10 choose 5 is correct. Any players are fair-game.

>> No.5546277

>>5546260
Giving it 2 more mins. If no solution by anyone, I'll post it with explanation.

>> No.5546279

A smooth, connected, compact 1-manifold (up to diffeomorphism)

>> No.5546281

Assuming no leading zeros are allowed, it's 9*10*10

>> No.5546285

>>5546275
Alright, since you're asking for ways to pick I guess it's correct. I answered the question "in how many ways can you make two equally sized groups out of the players".

>> No.5546287

>>5546277
>>5546281
This. 9 possibilities for first digit, 10 for second, 10 for third. Forth and fifth are second and first repeated.

>> No.5546288

>>5546270
>It stretches your mind
It really does.
I've encountered combinatorics in both Discrete Math and Statistics/Probability, and it's pounded me both times.
It's really cool, but like you said, it's challenging.
I might see it again before I graduate (2 classes away from math minor; CS major.)

>> No.5546289

>>5546277
9*10*10

9 because you can't start with 0.

>> No.5546293

>>5546285
Yes. That's correct.
>>5546281
Perfect.

Does everyone understand the solution? If not, I'll explain. Or, this gentleman can.

Here's one that's a little bit 'harder.' Or, one that I think is -- took me a few mins.

How many numbers greater than 3,000,000 can be formed by arrangements of 1, 2, 2, 4, 6, 6, 6?

>> No.5546298

>>5546293
4*6!=2880.

>> No.5546306

>>5546293
4*6!

>> No.5546308

>>5546298
I take that back.

>> No.5546307

>>5546293
Well actually, it didn't take me a few mins. Just a minute and a half after thinking about it. But, it was kind of interesting. Didn't encounter a problem like it before in the text.

Here's another one. Harder, for sure. How many ways are there to arrange MISSISSIPPI with no consecutive S's? Admittedly, I spent a LOT of time on it.

>> No.5546317
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5546317

>>5546298
>>5546306
>my face when

You..you sure about that?

>> No.5546333

Bumping and asking another question. This one is pretty hard, but I hope it involves something you all love to do. Well, I know I love to do it..because it reminds me of simpler times.

How many integer solutions are there to the equation x1 + x2 + x3 + x4 + x5 = 12 with xi greater than/equal to 0?

>> No.5546335

>>5546317
4*6! divided by 3!2!

>> No.5546350

>>5546317
Infinite. You never said how many digits we could use.

>> No.5546355

>>5546335
Not bad. Acceptable solution. Equals 240, which is indeed the answer. Here's my solution(and the one, coincidentally, which matched the back of the book).

So, we know that there are 7 spots in 3,000,000. But, of the numbers chosen, only 4 or 6 can go into that first slot where the '3' is. Hence, only 6 available spots are up for grabs.
Then, we're dealing with:
Permutation with 6 leading OR a permutation with 4 leading. But, by the addition principle, this means add. Hence,

6!/3!2!1! + 6!/2!2!

>> No.5546364

>>5546350
I did. Also, in your 'infinite' solution, you're ostensibly counting numbers 'less' than 3,000,000.

>> No.5546368

>>5546333
Welcome to DrRacket, version 5.3.2 [3m].
Language: Determine language from source; memory limit: 1024 MB.
> (require user/partition)
> (length (filter (λ(p) (<= (length p) 5)) (partition 12)))
47

>> No.5546369

>>5546307
>>5546333

These are still open problems. I have the solutions for each, so feel free to try and solve them. I can check them with my own numbers.

>> No.5546376

>>5546368
oh this is probably wrong because of reorderings

>> No.5546383

>>5545996
a one dimensional shape where a center can be found where all points are equidistant to it and infinite lines of symmetry.

>> No.5546384

>>5546333
16 choose 5?
Combinations with repititions, right? You can view the xi as containers into which you distribute 12 equivalent 1s.

>> No.5546385

>>5546368
>>5546376
Strict combinatorial logic only. No computer aids.

Two open solutions, gents.

>> No.5546389

>>5546383
WRONG! two dimensional

>> No.5546391

>>5546384
If only.

>> No.5546422

>>5546307
>>5546333
Giving it four more mins and then I'll expose the solution. They're not easy problems, especially the MISSISSIPPI one.

>> No.5546427

x^2+y^2=z^2

>> No.5546448

>>5546422
Okay. Guess no one has the solution. Here it is.

>How many arrangements of MISSISSIPPI are there with no consecutive S's?

First, let's get rid of the S's in the word and we're left with: M I I I I P P. Now, this is the tricky/interesting part. Place underlines before/after each letter. In other words,

_M_I_I_I_I_P_P_.

Now, those eight places are where the S's can go. So, out of 8 places, choose 4. Hence, 8 choose 4. Those are where S's need to go.

But now, M I I I I I P P can be permuted! Since the I's appear 4 times, the P's appear twice and the M once, plus there are 7 letters in total, we have: 7!/4!2!1!. The final answer is:

(8 choose 4)*7!/4!2!

>> No.5546450

>>5546391
Should be 16 choose 4.

>> No.5546456

>>5546448

>x1 + x2 + x3 + x4 + x5 = 12 with xi greater than/equal to 0?

This is actually a distribution/selection problem under the cloak. If you've studied those, then you know that '12' are the selected objects and you have n distinct boxes(or 5 in this case). Thus, there are:

16 choose 12 possible solutions. Pretty trippy. 12 +5-1 gives 16; the 12, you're picking each ones give 12.

Crazy.

>> No.5546465

>>5546450
16 choose 4 = 16!/12!4!

16 choose 12 = 16!/4!12!

Good stuff.

>> No.5546467

dr/dtheta = 0

>> No.5546549

>>5546206

It's such an old picture, I can't believe you'd actually fall for it.

God damn it, /sci/, get your shit together.

>> No.5546566

>>5545996
Of infinite points?

>> No.5546568

Doesn't it have to do with quartz? Like, isn't quartz the only material that can be made into a perfect sphere?

>> No.5546712

>>5546465
Where do you live?

>> No.5546740

>>5546568
>defining mathematical constructs with respect to the real world
pleb detected

>> No.5546742

>>5546712
New York.

>> No.5546747

>>5546568

you poor soul...

>> No.5546797

>>5545996

The set of points that are the same distance from a given point.

>> No.5546801

A circle is a line with a constant curvature.

>> No.5546803

>>5545996

r( theta ) = const.

/thread

>> No.5546804

>>5545996
>>5545996
>>5545996
>>5545996

Hey OP. Can't pull this one out of my ass but it has something to do with the integral of the equation of an ellipse, provided x = y.

>> No.5546810

>>5546797

In R^2 space

Otherwise a sphere would fit that definition

>> No.5546816

>>5546801
But what about a line with no curvature? that's not a circle, it's just a line.

>> No.5546820

>>5546810

>Otherwise a sphere would fit that definition

Why does that matter? A better example would be to point out that since the dimension isn't specified, any two points in R would be a "circle".

>> No.5546850

>>5546816

...or is it a circle with infinite radius?

>> No.5546887

isnt a circle just the collection of all points that are an equal distance from another point p

>> No.5547864

>>5545996
A line on the plane such that the furthest point on that line from any given point on the line is a finite constant, I think is the neatest intuitiony definition.

>> No.5547914

>>5546098

Equal sides is not enough, the angles should be the same too,

>> No.5547945

you take a line, and bend one end of it to the other end, and boom

>> No.5547954

>>5547945
a line is infinite

>> No.5547960

>>5547954
you take tenis ball, cut it in half, cover it in ink, and stamp it on a piece of paper, and there you have a circle

>> No.5547963

A circle is the set of all points in 2-dimensional space equidistant (by some distance r, the radius) from some fixed point C, the center.

What do I win?

>> No.5548035

Spec ( R[X,Y] / (X^2 + Y^2 -1) )

>> No.5548038

>>5546850
Projective geometry all up in his ass

>> No.5548150

let ||.|| be the euclidean norm on R^2, a an arbitrary point and r a positive real number.
the set {x \in R^2 : ||x-a|| = r} is a circle with radius r around the point a.
neat fact: you would get a square/karo if you would use the maximum norm / 1-norm