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.
In ##\mathbb{R}^2##, there are two lines passing through the origin that are perpendicular to each other. The orientation of one of the lines with respect to ##x##-axis is ##\psi \in [0, \pi]##, where ##\psi## is uniformly distributed in ##[0, \pi]##. Also, there are two vectors in...
I want to find the probability that the two points ($x_1, y_1$) and ($x_2, y_2$) lie on the opposite sides of a line passing through the origin $o = (0, 0)$ and makes an angle $\psi$ that is uniformly distributed in $ [0, \pi]$ with the $x$ axis when the angle is measured in clockwise direction...
Is the intersection of a 4D line segment and a 3D polyhedron in 4D a point in 4D, if they at all intersect? Intuitively, it looks like so. But I am not sure about it and how to prove it.
Hello there.Could coordinates be functions?For example in a n-manifold with (x1,...xn) let be the coordinates could they be functions of a coordinate system not belonging to the n-manifold?Or we could first use a coordinate system then have our results, and then have a second coordinate system...
Indeed, if we take a vector field which dual to the covector field formed by the gradient from a quadratic interval of an 8-dimensional space with a Euclidean metric, then the Lie algebra of linear vector fields orthogonal (in neutral metric) to this vector field is isomorphic to the...
Hi,
I take a big number of disks to composed a circle of a radius of 1 m, the blue curved line is in fact several very small disks:
I take a big number of disks to simplify the calculations, and I take the size of the disks very small in comparison of the radius of the circle. The center A1 of...
In the first sentence of Chapter 2 in Ben Crowell's "General Relativity" he states:
"The geometrical treatment of space, time, and gravity only requires as its basis the equivalence of inertial and gravitational mass".
This is stated as if it's an obvious fact, but I don't understand why. Why...
Hi PF!
I'm given a circle with parametric representation ##x=r\sin\theta,y=h+r\cos\theta##. There is also a line, which has the parametric equation ##x=x,y=\cot (\beta) x##. Note the line makes angle ##\beta## with the y-axis. When the circle intersects the line, it makes an angle, call this...
A recipient (cube) of 1m³ is filled of small spheres, there are for example 1000³ spheres inside the recipient. There are also 1000³ elastics that attract the spheres to the bottom. The elastic are always vertical. One elastic for each sphere. One end of the elastic is fixed on a sphere and the...
Could someone please write out or post a link to the Riemann Tensor written out solely in terms of the metric and its first and second derivatives--i.e. with the Christoffel symbol gammas and their first derivatives not explicitly appearing in the formula.
Thanks.
I can guess this question, by seeing that the surface area of the curved part must be in the form pi r l.
Don't know how to get to this formula though.
Answer is A
The $\triangle ABC$ and $\triangle AEF$ are in the same plane. Between them, the following conditions hold:
1. The midpoint of $AB$ is $E$.
2. The points $A,\,G$ and $F$ are on the same line.
3. There is a point $C$ at which $BG$ and $EF$ intersect.
4. $CE=1$ and $AC=AE=FG$.
Prove that if...
I was just thinking, if is said to me demonstrate any geometry statement, can i open the vector in its vector's coordinates? I will say more about:
For example, if is said to me: Proof the square's diagonals are orthogonal, how plausible is a proof like?:
d1 = Diagonal one = (a,b,c)
d2 =...
Could you provide recommendations for a good modern introductory textbook on differential geometry, geared towards physicists. I know physicists and mathematicians do mathematics differently and I would like to see how it is done by a physicists standard. I have heard Chris Ishams “Modern Diff...
Can anyone help me get started with this problem?
What should I use for Gni?
I've tried to produce Tni by working out Rni (using methods developed in an earlier chapter) but the results don't lead me anywhere.
I'm really stuck for a way forward on this problem so if anyone can help, it...
In convex quadrilateral $ADBE$, there is a point $C$ within $\triangle ABE$ such that $\angle EAD+\angle CAB=\angle EBD+\angle CBA=180^{\circ}$.
Prove that $\angle ADE=\angle BDC$.
Summary:: Suggest a geometry
Hello! I have difficulties with this question. It is translated from Swedish so if something's weird tell me.
The speakers in headphones often work with the help of magnetism, when a varying voltage is applied across a coil attached to the speaker membrane. The...
In "The Geometry of Minkowski Space in Terms of Hyperbolic Angles" by Chung, L'yi, & Chung in the Journal of the Korean Physical Society, Vol. 55, No. 6, December 2009, pp. 2323-2327 , the authors define an angle ϑ between the respective inertial planes of two observers in Minkowski space with...
Hey! 😊
Between the following two topics:
Elementary Geometry
Fibonacci and its sequences
which would you suggest for a presentation? Could you give me also some ideas what could we the structure of each topic? :unsure:
I am unsure how to go about this. I tried following the suggestion blindly and end up with with some cumbersome terms that are not the answer. From what I understand the derivative at each point would equal to T?
Answer: I just can seem to get to this. I think I'm there but can't get it in...
My context here is Finsler geometry with Cartan connection. I use ##x^\mu## for the usual spacetime position coordinates, and ##u^\mu \equiv \dot x^\mu## for velocity coordinates (the overdot denotes differentiation by an arbitrary parameter, not necessarily proper time). To explain the problem...
Ravi Vakil is teaching an online course in Algebraic Geometry over the summer due to COVID-19 for free.
It's based on his Foundations of Algebraic Geometry notes (for Math 216 at Stanford). It's starting soon. https://math216.wordpress.com/
Consider a point A outside of a line α. Α and α define a plane.Let us suppose that more than one lines parallels to α are passing through A. Then these lines are also parallels to each other; wrong because they all have common point A.
Summary:: Suggest a textbook
Good Morning
I have repeatedly tried to read Frankel's "Geometry of Physics" and I get swamped and overwhelmed.
(I hasten to add that as a MECHANICAL engineer, my math background has been deficient.)
I retire in about 10 years and I am looking to learn the...
CONTEXT: We are finding the the buoyancy force on a boat which is upright in a still water (Fluid at rest) and the only gravity is acting as the external force. So, first we go for imaging a proper geometry of our boat.
See this figure :
For this figure the book writes:
Fig 8 represents...
I always tend to get confused when thinking about non-Euclidean geometry and what straight lines and parallel lines are. If I think of a sphere, I get how two people driving north would almost mysteriously intersect at the North Pole and how the angles of a triangle would not add up to 180...
Can anyone derive the distance formula of a hyperbola for me, please? I have not found the derivation on the internet. I can't get any clue from the picture of hyperbola.
Summary:: We have a rotating arm, offset from the centre of rotation by a certain length, which is controlled by varying the length of a control rod. Need the angle of the rotating arm in terms of length of the rod.
The blue line is a fixed column structure. CE and BD form the rotational...
So far all I can work out is that the angle of incidence of the outer two and inner two rays is zero degrees, however, I can't work out how to get started on the problem. I feel like I need to use vertical slowness rather than the normal snell's law since I'm working with a dZ rather than a dX...
My partner asked me about questions no. 8 and 9.
Number 8 asks about what is the area of the quadrilateral.
Number 9 asks about what number is below the number 25.
Those are questions for Elementary School Math Olympiads in my country but both of us were having a hard time figuring them out...
Summary:: Suppose that [x, y] = e^{-3t} [-2, -1] is a solution to the system $x' = Ax$, where A is a matrix with constant entries. Which of the following must be true?
a. -3 is an eigenvalue of A.
b. [4, 2] is an eigenvector of A.
c. The trajectory of this solution in the phase plane with axes...
Hi
I have noticed that while I have the grasp of the theoretical underpinnings of linear algebra, I need work on applying it to geometric problems (think computer vision and rigid body motion). So, I am looking for a book that allows me to practice 3D geometry problems.
Is there any obvious...
I am uncertain if this belongs in the differential geometry thread because I don't know what area of mathematics my question belongs into begin with, but of the math threads on physics forums, this one seems like the most likely to be relevant.
I recently watched a video by PBS infinite series...
I'm trying to find angles α and β.
No additional information except: d, h, a.
I already tried to figure it out by using isosceles triangles, but this is only true when there is a equilibrium of forces. I thought there are similar triangles incorporated, but I get too many unknown variables...
Hello. If a closed ball is expanding in time would we say it's expanding or dilating in Riemannian geometry? better saying is I don't know which is which? and what is the function that explains the changes of coordinates of an arbitrary point on the sphere of the ball?
Hello. I am confused with this matter that how should we write the transformation matrix for an expanding space. consider a spacetime that is expading with a constant rate of a. now normally we scale the coordinates for expansion which makes the transformation matrix like this:
\begin{pmatrix}...
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...
Every explanation about scaling a 2D vector, or equivalently having a line segment PQ on cartesian plane and then find a point R on the line PQ satisfying PR/PQ = r (fixed given r) starts with that one specific case in the picture. A formula for the coordinates of R is then given for that case...
Is there ever an instance in differential geometry where two different metric tensors describing two completely different spaces manifolds can be used together in one meaningful equation or relation?
Hi, I just finished up with Riemann Geometry not to long ago, and did something with complex geometry on kahler manifolds. In your opinion what would be a next logical step for someone to study? I am very interested in manifold theory and differential geometry in general. I'm somewhat familiar...
Hello All,
I have yet another MCNP question. I received the following error "geometry error: no intersection found mcnp" when trying to run a a simulation. I looked at the output and according to it I have an infinite volume in cells 14 and 500. I plotted the geometry and don't see how its...
Hello everyone!
I have been looking for a general equation for any regular polygon and I have arrived at this equation:
$$\frac{nx^{2}}{4}tan(90-\frac{180}{n})$$
Where x is the side length and n the number of sides.
So I thought to myself "if the number of sides is increased as to almost look...
The metric for 2-sphere is $$ds^2 = dr^2 + R^2sin(r/R)d\theta^2$$
Is there an equation to describe the area of an triangle by using metric.
Note: I know the formulation by using the angles but I am asking for an equation by using only the metric.
https://en.wikipedia.org/wiki/Flatness_problem
The flatness problem (also known as the oldness problem) is a cosmological fine-tuning problem within the Big Bang model of the universe.
The fine-tuning problem of the last century was solved by introducing the theory of inflation which flattens...
Hi, I'm already familiar with differential forms and differential geometry ( I used multiple books on differential geometry and I love the dover book that is written by Guggenheimer. Also used one by an Ian Thorpe), and was wondering if anyone knew a good book on it's applications. Preferably...