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.
Homework Statement
Prove or disprove: Every translation is a product of two non-involutory rotations.
Homework EquationsThe Attempt at a Solution :[/B]
I am not sure if I got the right proof for the special situation: A translation is the product of two reflections with parallel reflections...
Currently I am face deep in three math books- my issued text for algebra, Schaum's Elementary Algebra, and Kiselev's Book I (Planimetry).
I love the stuff so far and want to be as knowledgeable as possible for when I can declare my major to Physics. One thing I know for sure is that algebra...
Show that y≈∆φ×secφ in the jpeg attached.
or ∆y = sec φ
A and B are points on curved surface, two lines are extended through origin to a line that is tangent to the circle, these points are A' and B', change in Angle will bring a change in length between A' and B'. I need to know how is this...
I posted the following on a phone forum and got no replies, as expected. It's a bit out-of-the-way and technical. Thought this might be a better place since the question still interests me.
I'm using my phone to take photos of assembly procedures on my electronics workbench. However, the focus...
Homework Statement
A quadrangular pyramid OABCD with square ABCD as the bottom. OA = 1, AB = 2, BC = 2 Also, OA perpendicular to AB, OA perpendicular to AD.
Question 1 : Find the inner product \overrightarrow {OA}.\overrightarrow {OB} and the size of the cross product |\overrightarrow...
In classical general relativity, gravity is simply a curvature of space-time.
But, a quantum field theory for a massless spin-2 graviton has as its classical limit, general relativity.
My question is about the topology of space-time in the hypothetical quantum field theory of a massless spin-2...
How much would time pass between watching the sun set from ground level and then watching it set again from the top of a sky scraper?
I heard once that this could be done using one of the towers of the World Trade Center. So I assume one could also do this using the Sears / Willis Tower in...
My goal is to do research in Machine Learning (ML) and Reinforcement Learning (RL) in particular.
The problem with my field is that it's hugely multidisciplinary and it's not entirely clear what one should study on the mathematical side apart from multivariable calculus, linear algebra...
In a thread that is now closed in the QM sub-forum someone wrote:
Isn't spacetime supposed to be fixed, "it just is", not changing or evolving?
To which I replied:
These days we know the core of GR - its simply this - no prior geometry (it's dynamical) which is the exact opposite of the...
Hi,
I am seeking to understand better how this well accepted idea:
"...according to general relativity, gravity is a manifestation of the geometry of spacetime."
(https://en.wikipedia.org/wiki/Loop_quantum_gravity)
is compatible with the equally well accepted idea that gravity travels at the...
The Problem is #16 in the attached picture. Essentially, I need to find the length of BC using information about congruency and the location of the centroid. I've been able to show a whole bunch of things, but nothing that gets me close to actually finding out the missing side length.
I began...
I am discussing physics with a friend and we need someone to confirm a thing that we're not agreeing on.
We are discussing incident light that is passing through different geometries, and I want to know how the light behaves when it reflects inside a half sphere (of glass for example). Maybe...
1. The problem statement, all variables, and given/known data
Triangle ABC has a point D on the line segment AB which cuts the segment in ratio AD : DB = 2 : 1.
Another point E is on the line segment BC, cutting it in ratio BE : EC = 1 : 4.
Point F is the intersection of the line segments AE and...
Hey all,
I was wondering if anyone had any good tips on debugging mcnp geometry? I'm an intermediate user working on better understanding the program. Does anyone have any tips or tricks that go beyond simply reading the manual?
Henry Wrinkler who played the Fonz in the 1970s sitcom Happy Days took geometry several times in high school. Henry said that he never understood the Pythagorean Theorem. In several recent and not so recent interviews, Henry Wrinkler aka THE FONZ talked about the fact that his parents put him...
Is it possible to have a magnetic field of a any geometry we want, or there are only few types of geometries that can be achieved with permanent magnets and electromagnets? If the former, how do we produce a magnetic field of specific geometry? For example, can the magnetic field be cylindrical...
Suppose if we have a cube:
The volume of the cube is the product of the length, width and the height. All this time, I've been looking at it as: To get the volume, multiply the area of the cross section of the cube by how many "layers" it has. To elaborate with the diagram given, one can see...
I was wondering if I read Kiselev's geometry books, would it count as a whole high school geometry curriculum? Currently, I am reading his first book, Planimetry, which is coming out as promising. I am planning to self-study geometry.
Homework Statement
Prove that any chord perpendicular to the diameter of a circle is bisected by the diameter.
Homework EquationsThe Attempt at a Solution
I was thinking that maybe I could form two triangles, show that these triangles are congruent, and then conclude that the two lengths of...
Homework Statement
It is not ordinary problem, it is connected to it. I don't understand a figure and how is described. The problem is here http://www.pmaweb.caltech.edu/Courses/ph136/yr2011/1102.2.K.pdf How did equation 2.4 from figure 2.3 arise? Why disappear ##\Delta t^2## from the right...
I am trying to make a spirally fluted pipe in ANSYS. I made the cross section as in the attached figure and sweeped on a straight line perpendicular to plane of the cross section. However I get a sort of a drill bit instead of the required spirally fluted pipe. Is there an issue with the cross...
Hi, all.
I would like to read books about the topics - Geometry, Real Analysis and Electricity and Magnetism. And I find the followings. Are they decent and rigorous?
Geometry
The Real Numbers and Real Analysis
Introduction to Electrodynamics
Classical Electricity and Magnetism
Electricity...
Hi, the question is from Serge Lang - Basic mathematics, Page 171 exercise 6.
Thing to prove:
Let ΔPQM and ΔP'Q'M' be right triangles whose right angles are at Q and Q', respectively. Assume that the corresponding legs have the same length:
d(P,Q)=d(P',Q')
d(Q,M)=d(Q',M')
Then the right...
I have a body in 3D-space and I would like to calculate the rotation axis when the body moves from A to B. I know the location (x, y and z) and the orientation (rx, ry and rz (axis angles)) at both A and B. The difference between A and B is small. The time instant during a dynamics simulation...
Hello Everybody!
Concept of Solid Angle was pretty much straight forward until they were on surface patches were taken into account which were visualized as base of cone.
I am having difficult when 3d Objects like Sphere/Cylinder .
We can very easily calculate the respective area and plugin the...
Connections between algebraic structures and geometries are mentioned in almost any course of modern geometry or algebra. There are monographs dedicated to the subject. Unfortunately, the books, I managed to find, are written for professional mathematicians. I am looking for a book that focuses...
Hi all,
According to the sunrise equation, the hour angle of the sun at sunset is:
cos H = -tan(a)tan(d)
where H = the hour angle, a = latitude and d = solar declination angle.
This equation says that H at sunset = -H at sunrise. Now, I have a few questions concerning that:
1) I was...
I came across this paper on the Geometry of M.C. Eschers work, thought I'd share the link. :smile:
http://www.math.cornell.edu/~mec/Winter2009/Mihai/section1.html
Homework Statement
Recall that in hyperbolic geometry the interior angle sum for any triangle is less than 180◦. Using this fact prove that it is impossible to have a rectangle in hyperbolic geometry.
Homework EquationsThe Attempt at a Solution
- I wanted to use the idea that rectangles are...
Hello, the name's Mike and I'm a newbie here,
I have a question pertaining to solar angles required to calculate a solar panel's hourly generation over one year. Total solar irradiance on a tilted surface equals the sum of the direct, diffuse and reflected component. In my case, reflection is...
Homework Statement
Question: An ant starts at one vertex of a solid cube with side of unity length. Calculate
the distance of the shortest route the ant can take to the furthest vertex
from the starting point.
Now, in the answer, the cuboid is unfolded and a rectangle of side 1 and 2 is...
1. Is GR about fixed curved background or dynamical...When Einstein first proposed GR.. did he mean it to have fixed curved background or dynamical?
2. As I understand it. At present. We treat GR as fixed curved background.. so that when we do QFT in curved spacetime, we fix the stress-energy...
Hi all,
I'm stuck on progressing a problem i have received some feedback around as detailed below. I would greatly appreciate some assistance, and thank you in advance for your time and contributions.
So i have a linear system:
Which is row reduced to:
I have identified that the system has...
Homework Statement
In Fano's Geometry, we have the following axioms a. There exists at least one line b. Every line has exactly three points on it c. Not all points are on the same line d. For two distinct points, there exists exactly one line on both of them e. Each two lines have at least one...
By the above question, I mean how would one effectively study synthetic geometry (geometry that makes no reference to explicit formulas or coordinate systems, like described in Euclid's Elements)? Do you just read through the propositions, try to reconstruct them later and perhaps more or are...
Hello!
I would like to know if anybody here knows if there's any good book on academic-level dfferential geometry(of curves and surfaces preferably) that emphasizes on geometrical intuition(visualization)?
For example, it would be great to have a technical textbook that explains the geometrical...
Good Day
Early on, in Frankel's text "The Geometry of Physics" (in the introductory note on differential forms, in fact, on page 3) he writes:
"We prefer the last expression with the components to the right of the basis vectors."
Well, I do sort of like this notation and after reading a bit...
I wanted to study General Relativity, but when I started with it, I found that I must know tensor analysis and Differential geometry as prequisites, along with multivariable calculus.
I already have books on tensors and multivariable calculus, but can anyone recommend me books on differential...
Homework Statement
Let ## (M, \omega_M) ## be a symplectic manifold, ## C \subset M ## a submanifold, ## f: C \to \mathbb{R} ## a smooth function. Show that ## L = \{ p \in T^* M: \pi_M(p) \in C, \forall v \in TC <p, v> = <df, v> \} ## is a langrangian submanifold. In other words, you have to...
Hi there guys,
Currently writing and comparing two separate Mathematica scripts which can be found here and also here. The first one I've slightly modified to suit my needs and the second one is meant to reproduce the same results.
Both scripts are attempting to simulate the trajectory of a...
Determine the area of the painted hexagon, knowing that the area of triangle ABC is 120cm^2
IMG Link: https://m.imgur.com/a/WtdsW
I tried using Heron´s formula, but just ended up with a bunch of terms and one more variable.
Sidenote: I guess part of it is figuring out that the side lenghts...
Lets assume we are mapping one face of earth. we place a plane touching the Earth at 0 lattitude and 0 longitude. Now we take the plane of projection. suppose that we expand the projection unevenly. The small projectional area of a certain lattitude and longitude is expanded by a factor which is...
Good morning (or evening),
I have a geometrical tricky question, which I need your assistance with.
Look at the following sketch:
In the sketch you see a pyramid ABCD. Inside the space of ABCD, you see a plane MKPN, where M, K, P and N are points on the pyramid sides.
Using the axioms of...
Yes, one more reason to be humble, I know. This is the simplest problem I couldn't solve so far.
Assume we have a circle of center O, a ruler of arbitrary size and a pencil.
We use the ruler and the pencil to choose 4 points on the circle - the extremities of two diametral/diagonal segments...
Homework Statement
This picture:
http://i.imgur.com/n015WjU.png
It's drawn with exactly the amount of information from the worksheet. Specifically, the two secants meet at a point, with an angle of 28 degrees between them. Both secants partition off an arc of 120 degrees. The goal is to...
So I was reading this book, "Euclidean and non Euclidean geometries" by Greenberg
I solved the first problems of the first chapter, and I would like to verify my solutions
1. Homework Statement
Homework Equations
[/B]
Um, none that I can think of?
The Attempt at a Solution
(1) Correct...