Euclidean Plane Puns

This is the current question I am working on. I am unsure on what I should do with this problem I keep trying to say "Let AB+CD=AG..." and working different ways with that but that is not working and I am unsure on where I should go. I been on this question for almost a week now and am ready to tear my hair out.

Dear Grant Sanderson

I appreciate you and your colleagues for sharing beautiful and useful videos. I would like to share with you a very interesting geometry problem as well. I'd seen this problem years ago and I remember that it has taken more than three months for me to solve it. Anyway, I hope you find the solution much faster and even you might want to make a video for it.

problem description:

A-- Draw a triangle of your choice(Attachment1).

B-- Draw angles of thirty degrees on each side of this triangle. In other words, make other outward triangles on each side of the main triangle. These three triangles are isosceles because they have two angles of thirty degrees and one angle of 120 degrees. They also have a common side with the main triangle(Attachment 2).

C-- Connect the outer vertices of these three triangles and create a new triangle. (Attachment 3)

D-- Prove that this new triangle will always be equilateral. It means that all three sides and all three angles of this triangle are always equal.

Faithfully Yours,

Mehran Kazeminia

Attachment1

Attachment 2

Attachment 3

I was looking through https://en.wikipedia.org/wiki/Euclidean_tilings_by_convex_regular_polygons this Wikipedia page, and i noticed that there was no tiling that used more than 4 regular convex polygons, namely the triangle,the square,the hexagon and the dodecagon. Is there one that uses at least 5 regular convex polygons? It doesn't have to use only regular polygons, although that would be nice.If not, then is there a finite number of ways you can tile the plane using a triangle, a square, a hexagon and a dodecagon?

Are there chaotic behaviors arising from geometric constructions? Say, we would specify some straightedge and compass construction rules, and observe how the resulting diagram looks like. We, of course, say that chaos occurs when changing some parameter (say, we deform a triangle slightly) might result in drastic shifts, be it "deterministic chaos" or otherwise. I don't believe chaos exists in Euclidean geometry, as we would have a couple of polynomial equations describing how the output diagram would turn out given the inputs, but then again, perhaps I'm in for a surprise!

Before looking at the proof of basic theorems in Euclidean plane geometry, I feel that I should draw pictures or use other physical objects to have some idea why the theorem must be true. After all, I should not just plainly play the "game of logic". And, it is from such observations in real life that result in the list of axioms. But I have great difficulty in doing so. In other words, I have great difficulty in understanding the natural behaviour in the real world that is described in the theorem. For example, below are two basic theorems from Euclidean plane geometry.

1. Given two lines cut by a transversal. If a pair of corresponding angles are congruent, then the lines are parallel.
2. (Basic proportionality theorem) If a line parallel to one side of a triangle intersects the other two sides in distinct points, then it cuts off segments which are proportional to these sides.(and its converse)

Another example,
3. the sequence of 1/n (n is a positive integer) converges to 0.

My thoughts about theorem 3, In contrary to theorem 3, we can want to have a number x that is less than 1/n for any positive integer n. Then, people in the past would have come up with a different standard list of axioms for real numbers. However, we do not want to have x in our real number system, hence leading to the question, "Why wouldn't we want such a number x to exist in our real number system?" I think, suppose (in real life) a physical object U representing the unit (i.e. 1), and no matter how small another physical object V is, we try to cut U into sufficient parts, whereby one part is smaller than V. But, can we, or no? If we can cut U into sufficient parts, whereby one part is smaller than V, then this idea leads to we should not have x in our real number system, and if we can't, then we will have the x in our real number system. My problem is, I can't decide on this.

What would you do for these 3 theorems? Feel free to use other basic theorems as examples in addition to these 3 theorems. What should I do to overcome my problem?

Given the standard graph with (-∞,∞) x axis and (-∞,∞) y axis, I'm trying to understand what each quadrant actually does. If I'm wrong, please correct me. Quadrant 1 represents any positive solution, given a positive domain, right? Quadrant 2 represents a positive solution as well, but with negative domains.. So, now does Quadrant 3 represent any negative solution when given a negative domain? If this is right, then Quadrant 4 will show any negative solution given a positive domain. Am I on the right track here? I'm getting into transformations, and want to have good conception about what's actually going on on the graph. All of my teachers were more algebraic and not as visual. I'd like to know what's actually going on.

One thing that has been on mind is "what happens to the Rogue's depth perception when he goes to the hyperbolic plane"? Like depth's will he perceive and such.

Anyways, I've mostly figured out the answer, but its kind of long.

Anyways, how are we going to go about this? Well, assuming both the rogue's eyes work, he is going to use stereopsis. That means that if his eyes can see a point p, his brain will use the angle of the eyes to calculate the location of p in relation to himself. Moreover, I am assuming that the rogue's brain will use Euclidean geometry to do this.

So, given the angles from the eyes to the point, finding the location p is just a matter of triangulation. If you form a triangle with a point, the first eye, and the second eye, we will call a the angle at the first eye in radians, and b the angle at the second eye in radians.

However, the hyperrogue is actually in hyperbolic geometry. What will the angles be?

Well, to simplify, will we only work in two dimensions. Stereopsis works in 2D, so this is fine. From the midpoint between the rogue's eyes, we will call d the distance to some point, and phi the angle. d will be absolute units, and phi will be in radians, with straight forward corresponding to 0.

The triangle formed by the first eye, the point, and the eye midpoint is solvable. That is because we know the distance from the point to the eye midpoint, the distance from the first eye to the eye midpoint, and the angle based at the eye midpoint, making this a solvable triangle. Therefore, we can calculate the angle at the first eye, a. We can do the same thing for b.

Therefore, if we know where a point in hyperbolic space is in relation to the rogue (and know the curvature of the space and distance between his eyes), we can figure out where his Euclidean brain will think the point is.

Well, being able to figure it out is nice, but what are the results? Well, I wrote up a program in Wolfram Notebook to calculate it. If you make a copy, you can play around with it yourself.

What are the results if you do not want to run itself?

Well, for one, the perceived depth will change if the rogue rotates his head. Taken to an extreme, this would probably make someone super nauseous, but luckily the amount it changes will be small if the distance between the eyes are small. With absolute distance 3 m, eye distance

The only real difference between countable and uncountable sets is the fact that uncountable sets have been proven to have no possible algorithm to arrange them into a linear order.

Tile a euclidian plane with squares, and it's a fairly simple proof that the tiles are countably infinite. Define one tile as the center tile, give each other tile a label based on how many tiles are between it and the center tile to create a countable series of rings, each with a countable number of tiles.

Similar methods can be used for any regular tiling of the euclidian plane. For tilings with multiple tiles, like the archemedian tilings, you can simply define a single repeating set as the center "supertile", use the underlying symmetry pattern to again define supertile rings, and you just have the extra step that each supertile has countable subtiles to label.

Where I'm lost is aperiodic tilings, like Penrose tiles. Is there a way to prove these aperiodic tilings are countably infinite as well?

I recently learned about the Euler characteristics of some surfaces, such as the sphere and the g-torus, in a combinatorics class. Since the plane and the sphere both have Euler characteristic 2, I was wondering if they are topologically homeomorphic.

I'm in my second year of high school algebra, so I'm by no means an expert. I'm towards the top of my class and math's my favorite class. I'm also interested in how this would be proven.

Edit: this question reduces to whether a square can be circumscribed onto any triangle. I've determined this to be possible for any obtuse, right or isosceles triangle. still unsure on an acute scalene, however.

The exact question reads as follows: Suppose each point in the Euclidean plane is colored either red or blue. Show that there exists two points of the same color that are distance 1 away from each other. Id like help on how to set up the proof, as the answer itself doesn't mean anything if i don't understand it. Discrete concepts are quite the challenge for me, so any insight into this would be much appreciated.

Hi,

I'm trying to find such a surface, so far ive got a cylinder of radius 1 because the first fundamental form has identity matrix. Is this correct? and are there any more?

thanks

The focus (for paraxial rays) is at the focus. But for another group of paraxial rays at a different angle or to a different pole its foci would be different. My question is whether the locus of the focii make a plane or a curved spherical plane?

I think it should be a spherical plane but teacher said it's not. Any proofs would be appreciated.

The article in Nature

Free PDF Mirror: http://www.scribd.com/fullscreen/72437812

Hi all,

My first post in /r/math. Honestly, this is not a homework problem. It is a part of a larger research project I am working on for which I need to prove this result. I am sure this has been proven elsewhere and would actually appreciate the reference, if anyone is aware, so that I can simply cite it and not reinvent the wheel.

I already tried to numerically verify this using excel and the result as such is true, but I am trying to formally prove that

[; \sqrt{(\lambda p + (1-\lambda) r - x)^2+(\lambda q + (1-\lambda) s - y)^2} \leq \lambda\sqrt{( p - x)^2 + (q-y)^2}+(1-\lambda)\sqrt{( r - x)^2 + (s-y)^2} ;]

where

[;0 \leq \lambda \leq 1;]

and

[;(p,q);] and [;(r,s);] are two points whose distance is being measured from given fixed point [;(x,y);] on the Euclidean plane.

What is the simplest proof of this?

Basically, prove that a vertical line intersects every horizontal line in the plane exactly once, no more and no less.

Edit: How I feel after making this post

So in the latest episode of Wonder Egg Priority one of the trauma's posed a question to Neiru and one of the egg girls. The question was to find the radius of the circle that circumscribes that intersects the points A, B, C, and D. Now this crazy looking professor dude claimed that it was impossible with a zero percent success rate...I immediately pressed X to doubt. The girls said that no such circle exists, but is that really the case? Well I will find the answer to that question.

The full written proof will be found in the following four photos:

1. Proof Part 1
2. Proof Part 2
3. Proof Part 3
4. Proof Part 4

Or hopefully this formatting works out and I can directly add the images:

First, as seen in proof part 1 we are given a generic non-representative look at a 4-gon (4-gon is a polygon with 4 sides, aka a quadrilateral) inscribed in a circle and we are given the length of each of the sides of the 4-gon. The length of the sides are: A_B = 7, B_C = 11, C_D = 11, and A_D = 9. This is all the information we need to solve this problem.

Solving this problem requires us to find out 3 things

1. Is there a circle that circumscribes points A, B, and C?
2. Where is point D located?
3. If there is a circle that circumscribes points triangle formed about points A, B, and C, will it also contain point D on the circle's perimeter?

So LET'S GET STARTED!!!!!

First let us establish the coordinate system that we will be working in. The most convenient coordinate system, I believe, is the one centered on point B with the central axes aligning with lines A_B and B_C because A_B and B_C are perpendicular. This makes the coordinates of points A, B and C:

A = (0,7)

B = (0,0)

C = (11,0)

Proof part 1

Next we move onto Proof Part 2 where we want to find the circle that intersects points A, B, and C. The stand equation for a circle is as follows:

(x-h)^2 + (y-k)^2 = r^2 (or)--> x^2 - 2hx + h^2 + y^2 - 2ky + y^2 - r^2 = 0 (1)

Where h and k are constants t

{{Trip report written on November 25, 2020. Apologies for my rough formatting, time dilation was too strong to give an accurate time frame}}

Last night, I consumed 888mg DXM Polistirex, roughly 1-1.5 grams of psilocybin cubensis, and cannabis in the form of bong hits, a joint, and dabs.

I only purchased the DXM Polistirex because there was no hydrobromide in stock at my local pharmacy. In general, I prefer the hydrobromide experience as it is easier to keep track of. There’s no silly 8-12 hour time frame you have to guess around; that is, my experiences show that hydrobromide produces experiences with much more consistent durations.

I have taken this specific batch of cubensis before -- they are very potent but seem to fluctuate blood sugar quite heavily. I would feel a continual come-up and acquisition of energy, to the point where I would get a bit shaky, and then it would all disappear. When my blood sugar would drop, it felt like some essential force within reality was drained from my existence, and I was fading away (both physically and spiritually). I would get the sense that I was about to faint. I can only presume that it is during the peak that this happens, and it seems to be exacerbated by cannabis in my particular case. (Update, after many months: this was probably mental. I’m very prone to somatic anxiety. While I haven’t tried mushrooms since this experience, I am sure if I were to, I would not experience the same fainting episodes.)

Nonetheless, I found myself with a completely free day on a week-long vacation from my day to day life. In terms of set and setting, I should note that I have been feeling well recently. I completed several large tasks in my life that have brought me a sense of accomplishment. For the setting, I had my house with my two cats. I decided to dose.

I should note that I made sure to feed myself a balanced meal before this experience. Not too large to the point where I would feel nausea from the DXM, but enough to propel me through the mushrooms without feeling woozy.

I dosed roughly 1 hour and 50 minutes after I ate this meal, at 7:30 PM. I chugged a 5-ounce orange-flavored bottle of Delsym and washed it down with an apricot La Croix. Using similar flavored water to wash down the syrup worked pretty well, surprisingly.

I decide to smoke a small bowl 15 minutes after I consume the syrup. In moments, I am already finding myself relaxed. A bit disconnected. I tend to be overly sensitive towards my bodily sensation

Here's the 17th set of 10 questions in the 260 question quiz. It's unfortunately Q, which is almost as much fun to write as X.

Sticky thread for info and scores.

All answers begin with Q and are in alphabetical order. The answers are in spoilers after the questions.

Post your score in the comments and I'll add you to the scoreboard. Good luck! :)

Questions:

1. A polygon in Euclidean plane geometry with four edges (sides) and four vertices (corners) would be better known as a what?
2. From the Italian for '40 days', what name is given to an isolation period for people or animals with a contagious disease?
3. Founded in 1608 as the capital of New France, what is Canada's oldest city?
4. Which British band released two hit albums in the 1970's with titles taken from Marx Bros films?
5. What is the French name for a tart of cheese and bacon in a cream and egg filling?
6. By what name is "Calcium Oxide" more commonly known?
7. Native to the Middle East and central Asia, which small tree bears a bitter, yellow, pear-shaped fruit?
8. Tonic water contains Which drug, which can be extracted from the bark of the cinchona tree?
9. Lying on the slopes of the volcano Pichincha, what is the capital of Ecuador?
10. "The minimum number of members of a deliberative assembly necessary to conduct the business of that group." A what?

Answers:

1. >!Quadrilateral##!<
2. >!Quarantine###!<
3. >!Quebec######!<
4. >!Queen#######!<
5. >!Quiche Lorraine!<
6. >!Quicklime####!<
7. >!Quince######!<
8. >!Quinine######!<
9. >!Quito#######!<
10. >!Quorum#####!<

Cheeky website link...

Just today, Yuchen Liu, Chenyang Xu, and Ziquan Zhuang put up a preprint solving the so-called finite generation conjecture, a conjecture in algebraic geometry that forms the last link in a long chain of conjectures in the study of the K-stability of Fano varieties, a huge topic of research in algebraic geometry over the last several decades. Since the resolution of this conjecture essentially completes this field of study, I thought it would be a good idea to post a reasonably broad discussion of it and its significance.

In this post I will summarise this research program and the significance of the paper, and where people in the field will likely turn to next.

##Introduction

Going all the way back to the beginning, the problem starts with what pure mathematicians actually want to do with themselves. The way I like to think of it is this: pure mathematicians want to find mathematical structures, understand their properties, understand the links between them, and classify them (that is, completely understand which objects can exist and hopefully what they all look like). Each of these is an important part of the pure mathematical process, but it is the last one is in some sense the "end" of a given theory, and what I will focus on.

In geometry, classification is an old and interesting problem, going back to Euclid's elements, where the Platonic solids (Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron) were completely classified. This is a fantastic classification: pick a class of geometric structures (convex regular 3-dimensional polytopes) and produce a comprehensive list (there are 5, and here is how to construct them...). Another great classification is the classification of closed oriented surfaces up to homeomorphism/diffeomorphism. For each non-negative integer g called the genus, we associated a surface with g holes in it.

##Higher dimensional classification

As you pass to more complicated geometric structures and higher dimensions, the issue becomes more complicated, for a variety of reasons. Perhaps the most obvious is that classification of all geometric structures is impossible. This is meant in a precise sense: a classification should be some kind of list or rule which can produce all possible structures of a given type. However it can be proven that every finitely presented group appears as the

Let there be two not-intersecting ellipses in the standard Euclidean plane. You can draw lines that are tangent to each ellipse. In particular, there are four lines that are tangent to both ellipses at the same time. Find their slopes and tangency points.

I'm aware of Habert's paper on the subject, but the computation is way too complicated for my tastes. That is, he uses polynomials, and depends on Maples solver to get solutions to bitangents equations. And it just looks like messy analytic geometry.

I'm looking to get a more geometric construction for a project I'm programming. Something like: rotate, translate and scale the two ellipses so that they're in some standard-looking situation (an x-axis aligned ellipse with a circle somewhere around it seems like a reasonable situation), find the lines, then do the basic transforms in reverse.

Inverse trig is allowed. Anything to make this work reasonably well and be easy to explain.

Are there any existing resources for this exact problem? If not, are there good resources for "nice" modern geometry constructions for ellipses and their tangents? It doesn't have to be compass and straightedge; I don't think that's possible, actually.

I'm eager to learn more advanced planar geometry, as I don't think I learned this stuff even in my university's pure math program...

So, I started reading this article about different theories regarding what black holes actually are, and as is the case with most articles like that, it raised more questions than it answered.

First of all, if nothing smaller than the Planck scale is relevant and can't be said to exist at all, aside from some weirdness I seem to remember Brain Greene writing about with regard to string theory, what is the difference between a singularity and a Planck Length, Area, or Volume? It seems to me that they are by definition synonymous. The article only mentions the Planck Length, so I don't know if they really mean a one dimensional, string like core inside a black hole, which in this case isn't really a black hole at all, or if they mean a Planck Volume.

Secondly, what the hell is a Planck Area or Volume anyway? I've always envisioned a square and cube respectively, but with regards to a black hole it seems to me that if the center is really one Planck Volume worth of matter that it would have to be spherical. And got me thinking that one Planck Length^3 is actually larger than a Planck Volume since all but two planes are wider than a Planck Area. Likewise, a Planck Area has only two lines that are actually a Planck Length, so wouldn't it really be a circle? And if it is truly a square, how does that square square (sorry) with the idea that a black hole's entropy can be measured in terms of the number of Planck Areas on it's event horizon when they would necessarily have to be (non-euclidean?) squares with angles greater than 90°? It doesn't make sense to me if any part of the Planck Area or Volume has a dimension longer than a Planck Length, which doesn't seem to ever be possible in a curved spacetime.

Lastly, and this relates to my first question, why would a black hole cease to be a black hole and cease to have an event horizon just because the heart of it turned out to have non-zero dimensionality? From what I remember, it was theorized way back when that were a star to grow large enough that it's escape velocity exceeded C, it would be dark and therefore have an event horizon, no singularities necessary. Obviously now we know that's not exactly how it works, but nevertheless if the star was still there behind the veil of the infinite curvature it creates, how is that any different from a singularity? Either way, there's no way out aside from Hawking Radiation so it seem

{{Trip report written on November 25, 2020}}

Last night, I consumed 888mg DXM Polistirex, roughly 1-1.5 grams of psilocybin cubensis, and cannabis in the form of bong hits, a joint, and dabs.

I only purchased the DXM Polistirex because there was no hydrobromide in stock at my local pharmacy. In general, I prefer the hydrobromide experience as it is easier to keep track of. There’s no silly 8-12 hour time frame you have to guess around; that is, my experiences show that hydrobromide produces experiences with much more consistent durations.

I have taken this specific batch of cubensis before -- they are very potent but seem to fluctuate blood sugar quite heavily. I would feel a continual come-up and acquisition of energy, to the point where I would get a bit shaky, and then it would all disappear. When my blood sugar would drop, it felt like some essential force within reality was drained from my existence, and I was fading away (both physically and spiritually). I would get the sense that I was about to faint. I can only presume that it is during the peak that this happens, and it seems to be exacerbated by cannabis in my particular case. (Update, after many months: this was probably mental. I’m very prone to somatic anxiety. While I haven’t tried mushrooms since this experience, I am sure if I were to, I would not experience the same fainting episodes.)

Nonetheless, I found myself with a completely free day on a week-long vacation from my day to day life. In terms of set and setting, I should note that I have been feeling well recently. I completed several large tasks in my life that have brought me a sense of accomplishment. For the setting, I had my house with my two cats. I decided to dose.

I should note that I made sure to feed myself a balanced meal before this experience. Not too large to the point where I would feel nausea from the DXM, but enough to propel me through the mushrooms without feeling woozy.

I dosed roughly 1 hour and 50 minutes after I ate this meal, at 7:30 PM. I chugged a 5-ounce orange-flavored bottle of Delsym and washed it down with an apricot La Croix. Using similar flavored water to wash down the syrup worked pretty well, surprisingly.

I decide to smoke a small bowl 15 minutes after I consume the syrup. In moments, I am already finding myself relaxed. A bit disconnected. I tend to be overly sensitive towards my bodily sensations, so it was entirely possible I was experiencing placebo effects from the DXM, but I am unsure.