Geometry Definition and 999 Threads

I love physics and do it as a hobby. I also like to use AI to help me make new and exciting discoveries. I look forward to getting feedback for some of the papers I wrote. All feedback is welcome. All I ask is to please be friendly. I am looking forward to having some exciting conversations.

I had two rather long questions about the recent programe on searching for UV-completions of EFTs through positivity bounds (that is, UV completions that obey fundamental constraints given by QFTs: unitarity, locality, causality, analyticity and Lorentz invariance) Question #1: The EFT-hedron...

I want to prove that ##\mathcal{O}_{\mathbb{A}^n}(\mathbb{A}^n \setminus \{0\}) = k[X_1, \ldots, X_n]## for ##n \ge 2## and some user on another forum gave this proof: "We know that ##\mathcal{O}_{\mathbb{A}^n}(\mathbb{A}^n \setminus \{0\}) = \bigcap_{i = 1}^n \mathcal{O}(D(X_i))##, where...

I have made an attempt, which I give below. There are two issues. The first is that I am missing a factor of 2 in the MOI of the sphere. The second issue is that the integral I am getting for the spherical shell is very complex. I have tried symbolab and it chucks out some horrible things. This...

source of question: https://www.tes.com/teaching-resource/vectors-harder-gcse-practice-questions-for-edexcel-1ma1-syllabus-12621082 Found this vector question by a google search, apparently a person has made this question and it seems almost impossible to solve it without using geometry. Even...

if 2d = mobius strip 3d = klien bottle what could 4d be??

I thought it would be fun to start a thread where we can share and discuss interesting mathematical puzzles. Whether you have a puzzle you've been pondering or a favorite one you'd like to challenge others with, feel free to post it here. Let's see how many we can solve together! Here's one to...

I am a retired lawyer who has been pondering tiny 5th dimensions since I first heard about them in 1986. I think I sort of understand the Kaluza-Klein theory in a geometrical sense. I'm trying to connect the dots of what I think I understand to what the standard model predicts. Does anybody here...

I've got a small geometry related error in my MCNP input deck, corresponding to cell #14 (the outer edge of my detector model). This should be a quick fix, but I'm running into issues defining that particular union of surfaces. Any assistance is appreciated.

I can find radius of incircle of ADE and incircle of ABC triangle but I don't know how them help me.

Suppose we have a 2-sphere, and an associated metric. For specificity $$d\theta^2 + \sin^2 \theta d\phi^2$$ On this 2-sphere, lets consider two different connections. The Levi-Civita connection, where geodesics are great circles, and a connection with torsion where the curves of constant...

In a study I did in MCNP, I want to create a sliced geometry in a torus structure. But MCNP cannot calculate the volume of the geometry I created. What should I do about this issue? 2 0 -50 -12 13 3 0 #2 12 pz 34.4985 13 pz -34.4985 50...

I was reading a book on complex numbers where I stumbled upon this article. I was following it when I found some error. The distance between the center of circle and line should be mod(z star - z1) right? Z0 is just any random pt on line. They have continued this fallacy in example 3.10...

Hello All, I am an 8th grader taking honors geometry at the high school. I am accelerated in math by one year, but it quickly got boring and I wanted to do honors algebra II. I took the test for geometry (midterm and final) and failed to pass. This was about 4 months ago. I am trying to double...

If I'm given the coordinates (edit: in a plane) of 4 vertices of a quadrilateral, which conditions should I check to see if it is a square? Will it be sufficient if I check if - all 4 sides are equal all 4 angles are right angles (or even 2 angles are right angles?) or are there other...

I'm trying to rotate an RPP 45 degrees around the y axis (BUT NOT THE ORIGIN Y AXIS, rather the y axis at x=a, z=b). Is there a way to do this in MCNP? I've tried every single possible combination of angles and inputs to no avail. Again, I have an RPP that is not centered at the origin and I...

I'm looking for someone to help troubleshoot my MCNP input deck. I'm getting a geometry related error most likely due to some sort of surface overlap. Haven't been able to identify the issue myself, so I'd appreciate a secondary check. Thanks!

We are trying to control a container with water inside. The container has two phases of operation. In one phase, the container is tilted. In this tilted state, the distance from the observation point (red dot) to the surface of the water (distance B) is known from a measurement device. However...

TL;DR Summary: When asked to show the LOCUS OF ALL POINTS how precise should the questioner be? I was asked to draw THE LOCUS OF ALL POINTS 3cm from a line. I measured 3cm above and below the line, and drew two parallel lines there. Then drew two semi-circles with radii 3cm out from either...

Is there something about the repulsion of the lone pair on thenitrogen that prevents the cyanides being all on the same plane?

In geometry, there is a theorem pertaining to whether given an angle, side, and side gives 0, 1, or 2 triangles. See figure. In the figure, if x = 10sin(30) then there is exactly 1 triangle, if x > 10sin(30) then 2 triangles if x < 10sin(30) then no triangles. I think this has a theorem name or...

I try to build this geometry in mcnp, but I'm having difficulties. Can anyone help me? I'm having a lot of trouble defining these curves. A small introduction will help me a lot.

Hello! I wish to understand which lines and vertices in different 2D orthographic views of a 3D object correspond to each other. This information would also later be used to construct a 3D model from the 2D orthographic views. Blue shows matched edges/lines. Orange shows matched...

Firstly im not sure if the new triangle that we make of height 3 times the original height of an equilateral triangle of side a, will be an equilateral triangle as well or not. I assumed it would be, please let me know if I interpreted the question wrongly. Continuing this train of thought I...

I have a problem that I imagine does not have a closed-form solution and requires the use of some kind of optimization solver. I am not an engineer myself, so forgive me if the question seems stupid. The problem is as follows: I have a circle bound in a square, and an arm going from the center...

Let R = radius of big circle and r = radius of small circle. I worked out that ##R \times r=144## but then not sure how to proceed. I also know AB = CD = 24 cm. Thanks

Here is my sketch: The triangle ##abc## is arbitrary, the triangles ##acp##, ##abq##, and ##bcr## are equilateral with centroids ##m##, ##n##, and ##k##. I suspect that the triangle ##mnk## is equilateral. Here is my proof. By the equation for centroids, ##3m=a+c+p## ##3n=a+q+b## ##3k=b+c+r##...

Something I never would have suspected.

It's extremely hard, if not outright impossible, for our limited brain to visualize 4D objects. 3D objects are fine, but 4D is just impossible (except, maybe, to some people). However, at one point I figured out a way to comprehend (in some ways "visualize") the 4-dimensional hypercube, which...

Please excuse my ignorance I hear some people say that space is flat. What exactly do they mean by this?

If big bang existed, than universe must be sphere, because explosion expand in all direction.. I read that universe is maybe flat so how this is possible? Why can we with telescope determine shape of universe?

in what geometry would a cube of circles be possible

I'm trying to solve this for a model I'm making in OpenSCAD. Given a circle of radius r centered on the origin, and two perpendicular lines at x=a and y=b, where is the center (x1,y1) of a circle that is tangent to both lines and the centered circle? Here's a picture: I thought it would be...

I'm confused about what we are really measuring when taking the dot product of two vectors. When we say we are measure "how much one vector points in the direction of the other", that description is not clear. At first I thought it meant how much of a shadow one vector casts on another and I...

The converse of the supporting hyperplane theorem states Here's the "proof": I've been told that any proof that does not use the fact that ##C## has non-empty interior will not work, because it easy to construct counterexamples of sets that will fail if they have empty interior. I'm not sure...

In my recent study of Conic Sections, I have come across several proofs (many of those comprise Dandelin spheres) showing that the cross-section of a cone indeed follows the focus-directrix property: "For a section of a cone, the distance from a fixed point (the focus) is proportional to the...

As an explanation to the Cherenkov angle, images such as this are offered: This is used to explain the Cherenkov angle θ at which the Cherenkov radiation appears to be propagating. To figure this angle out however one has to assume that the wavefronts are tangent to each of these circles, so...

Could you please help me understand why the answer picture says the length of the two sides of the triangle are 1.33 yard each? Thanks

I have proved that triangles around the equilateral triangle are congruent, but I don't know how to prove that they are arranged in such a way that they actually do form an equilateral triangle. Like, do I write that they do form an equilateral because it's given in the problem that triangles...

Fascinating, and utterly unintuitive. This is a question that appeared on the American SAT test until it was recently removed. (citation: Veritasium, to which I will not link at this time.) Every student ever has gotten it wrong, and that's because the SAT writers got it wrong too. The...

I think that there might be several solutions. I drawed one possible situation: I think that this is just geometry, but I don't know how to solve it simply. I had an idea that if the beam was going through the black axis, then it would be easy to calculate, and that would be aslo solution for...

I can't find the angle between F(Big arrow in my drawing) and the y-axis! I am terrible in geometry, what book do you suggest to me as Mechanical Engineer to get better at geometry?? This is my attempt to solution and the figure:

TL;DR Summary: . An electrone moves in a magnetic field ##B(\vec r)=g \frac {\vec r}{|\vec r|^3}##. Why does the conservation of the quantity $$\vec J=\vec r \times\vec p +eg\frac {\vec r}{|\vec r|}$$ mean that the motion is on the surface of a cone?

Physicist Nima Arkani-Hamed has taken an approach to understand fundamental physics based on geometry (specifically, positive geometry). This started with his work with Jaroslav Trnka in the amplituhedron [1] and later it was generalised to the associahedron [2],the EFT-hedron [3]... I was...

Hi, this is a question about an article in the Scientific American magazine. In 1981 Bernstein and Phillips wrote an article about fiber bundles and quantum fields, and I believe it's still a useful reference, the kind of thing lecturers would use at university. Anyway, my question is, how do...

So I'm playing with this visualization from this other thread and I'm brute-forcing the "days" scale because don't really know how to place the marks. (by brute-forcing, I mean I am using SUVAT to calculate the distance one can travel in one day, then redoing it to calc the distance in two...

I actually do not understand where to place this thread. Hope that it is a high school level problem. There are two triangles ABC and PQR. The vertex A is a middle of the side QR. The vertex P is a middle of the side BC. The line QR is a bisector of the angle BAC. The line BC is a bisector of...

Can someone provide me with some free resources (classes, books, notes, sites anything) for co-ordinator geometry? I want to study it from the basics while understanding the logic of every step and build upto start of collage level. Note ; non free resources are welcome too, but free resources...