Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.
Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Gauss' Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in an Euclidean space. This implies that surfaces can be studied intrinsically, that is as stand alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry.
Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry.
Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, etc.
Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.
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...
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##...
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...
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?
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...
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...
Can any one please explain if I want to run mcnp 5 input file of certain geometry for flux calculation on different surfaces. So far as I know If I increase the NPS (number of particles) it wll give more accurate result but when I increase NPS from 10e9, input file do not run and close within a...
I find it interesting that so many know how to use all kinds of apps on their cell phones, but so few are able to do simple algebra any more. If you ask around, engineers not included, I think you would find very few people e.g. to be able to find the axis of symmetry of the parabola ##...
(a) The hint from question is to used geometrical argument. From the graph, I can see ##r_1+r_2=c_2-c_1## but I doubt it will be usefule since the limit is ##\frac{r_2}{r_1} \rightarrow 1##, not in term of ##c##.
I also tried to calculate the limit directly (not using geometrical argument at...
Gave following to bing chat and chatgpt, Both gave wrong answers. Any thoughts?
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Geometry problem: Semi-circle inside triangle: Triangle with known length sides a, b, c where a is the longest. Place inside the triangle a semi-circle with entire diameter resting...
Can anyone explain the meaning behind a mobius strip? Basically just a means to travel on both sides of a flat surface? It's still a 3D object though since it uses 3D space for the twist to be possible?
Can AI or GPT-4 answer the following?
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Geometry problem: Triangle with known length sides a, b, c where a is the longest. Place inside the triangle a semi-circle with diameter on side a. What is radius of largest possible semi-circle in terms of side lengths?
This is an Argand diagram showing the first 40,000 terms of the series form of the Riemann Zeta function, for the argument ##\sigma + i t = 1/2 + 62854.13 \thinspace i##
The blue lines are the first 100 (or so) terms, and the rest of the terms are in red. The plot shows a kind of approximate...
In the figure, the point S is located inside the section FE.
Starting from S, as indicated in the figure, six circular arcs are drawn step by step around
arcs around A, C, B, A, C, B are drawn.
Show that the sixth arc leads back to S and that the six arcs together are then exactly as long as the...
Hey
Condition 1:
A 2D infinite plane and there is a circular hole in the middle. When t=0, an impulsive loading, P=f(t), is applied to the boundary of the circle(outward), so the wave will start at the boundary of the circle and propagate in the plane
Condition 2:
A 3D infinite plane and there...
So I've got the following problem:
I have points A, B, and C which form a triangle in a 3D space (each point of the triangle has x,y, and z coordinates). I need to find out on which side of the triangle point D lies. I do not have access to the normal of the triangle.
How am I supposed to...
TL;DR Summary: Looking for books similar to "The Wonder Book of Geometry" by David Acheson
I loved David Acheson's "The Wonder Book of Geometry". Can you recommend other books like that?
Hi, I'm differentiating the "z" function to find extreme points but after solving the first partial derivatives with respect to "x" and "y" and also the "x" variable of the system, I can't find "y" (still in the system) using "ln" (natural logarithm).
The question is can I differentiate both...
This is the textbook question. I do not have the solution. I am pretty stuck on this one:cry:
My attempt on this...find my rough sketch here;
From my analysis;
##x+x+m+m=180^0## angles opposite each other on a cyclic quadrilateral... I have point ##O## as the centre of the circle...
Text question is here and solution;
My approach;
##BP ×AP= PT^2##
Let ##AP= x##
Therefore, ##(6+x)x=16##
##x^2+6x-16=0##
##x=2## or##x=-8##
##⇒x=2## positive value only.
I guess this may be the only approach. Cheers!
As I understand it, the flatness problem of Bob Dicke, says a flat universe in unstable and so has to be set very precisely in the early universe to give us the flat universe we see today. Is this the same problem as saying the expansion rate had to be finely tuned and if so how are the two...