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.
So I was reading now about the new geometries and I wanted to know if I can study the Reimann Geometry knowing that I finished high school or if I could just know about it but not about the formulas. I am so interested in the subject because it is used in astronomy.
I have found this paper on the internet and think it might be interesting for some on this forum because there are frequently questions similar to the ones the paper tries to answer.
http://arxiv.org/abs/1605.00890
http://arxiv.org/pdf/1605.00890v1.pdf
Homework Statement
Hello, my friend asked my If I could help him with this problem. However I just can't seem to find a way to solve this.
Ellipse
Focus(2,2)
vertex(2,-6)
Point(26/5,2)
a+e=8
find the equation of the ellipse
Homework Equations
(x-m)^2/a^2+(y-n)^2/b^2=1
Center(m,n)
a=moyor axis...
Homework Statement
Given that the sum of interior angle measures of a triangle in hyperbolic geometry must be less than 180 degree's, what can we say about the sum of the interior angle measures of a hyperbolic n-gon?
Homework EquationsThe Attempt at a Solution
So in normal geometry an n-gon...
Hey PF! I'm going through a textbook right now and it just said "obviously, you can't have an equilateral pentagon with 4 right angles in spherical geometry (Lambert quadrilaterals).
However, I am not able to make the connection. can somebody help me understand why this is?
I am planning on doing a huge self-studying over geometry and hopefully a little bit a trigonometry session over the summer at my public library. Over at a friends house, I see that he has a book about "algebra 2 for dummies" in his pile of science books and it got me wondering, are the "For...
Homework Statement
On the picture, ##ABCD## is a parallelogram, ##(EF) // (AB) ##, and ##(GH) // (BC)##.
The problem is : show that lines ##(EB)##, ##(HD)##, and ##(IC)## either all meet in ##M##, or are parallel.
Homework EquationsThe Attempt at a Solution
I've solved the problem...
Homework Statement
If ##α, β, γ, δ## are four complex numbers such that ##\dfrac{γ}{δ}## is real and ##αδ - βγ ≠ 0##, then ##z = \dfrac{α + βt}{γ + δt} , t \in ℝ## represents a
(A) circle
(B) parabola
(C) ellipse
(D) straight line
Homework EquationsThe Attempt at a Solution
Eqn of circle is...
Can you suggest a book that discusses properties of triangles and circles? (Like properties and theorems on circumcircle, excircle, nine point circle, etc).
Most of the geometry books are either to basic or too advanced. I have read a book on complex numbers by Liang Shin Hahn. But the...
Mod note: Member warned that homework questions must be posted in the Homework & Coursework sections
http://imgur.com/zGB2dnY
Was given this problem a few weeks ago and I'm not sure how I should be approaching it. Please let me know which theorems I should look into in order to solve the problem.
Homework Statement
In the attached diagram,the edge of the square is a. find the area of the shaded region
Homework Equations
area of circle = πr^2 ,area of square,triangles
(Please avoid using integration/radian angle/tangent...Since this problem is found in a maths exercise suitable for a...
Homework Statement
In a methane molecule, determine the length of the distance between a hydrogen atom at A and the carbon atom at O (see diagram) in terms of the length of the edge (e) of the cube at four of whose corners the hydrogen atoms rest.
Homework Equations
pythagorean theorem...
In discussing proportions (a topic to which I have not been properly exposed) Legendre states that, adding the antecedent of a proportion to the consequent, and comparing the sum to the antecedent, one obtains a proportion equal to the original plus unity. Legendre's book is apparently...
Hi
I'm an advanced undergraduate physics student and I'm currently searching books for my career project.
The topic I selected is titled: Algebra and geometry in modern physics.
So I'm currently looking for books that cover modern aspects of physics in a more mathematical approx.
In specific I'm...
Homework Statement
A piece of wire, 100 cm long, needs to be bent to form a rectangle. Determine the dimensions of a rectangle with the maximum area.
Homework Equations
P = 2(l+w)
A = lw
The Attempt at a Solution
This is what I don't understand, the solutions that I saw from looking around...
Hello all,
I am not too experienced with geometry. I am just curious whether it would be possible to define a shape based on variables.
Say you have a simple relationship between volume and some variables. V=x+y. This tells you about the volume of a 3D object, however, it does not describe the...
Hello,
A friend of mine gave me this puzzle and I'd like to share it with you, math enthusiasts:
Two ladders intersect in a point O, the first ladder is 3m long and the second one 2m. O is 1m from the ground, that is AC = 2, BD = 3 and OE = 1 (see the image bellow)
Question: what's the value...
In the question they say a "1/2 by 2 in rectangular cross section". Does this mean the dimension of 1/2 is actually into the page and not as shown in the diagram?
Also, In the solution there is mention of "the base metal adjacent to the weld" and "shank of the attachment". What areas are these...
Hello all!
I've just started to study general relativity and I'm a bit confused about dual basis vectors.
If we have a vector space \textbf{V} and a basis \{\textbf{e}_i\}, I can define a dual basis \{\omega^i\} in \textbf{V}^* such that: \omega^i(\textbf{e}_j) = \delta^i_j But in some pdf and...
Imagine a planet similar to Earth, but exposed to a completely different star. The star has the same mass and emits the same amount of photons as the sun, but it is a huge, extremely slender torus made of 1 mm diameter neon tubing. The planet is on the axis of the torus and at the same distance...
I would appreciate any help with this questions because I truly horrid at geometry questions.
I've only done (i) to which I've found the answer to be (E). I can't do from from part (ii) on.
Homework Statement
Given points of a triangle: A(4,1,-2),B(2,0,0),C(-2,3,-5). Line p contains point B, is orthogonal to \overline{AC}, and is coplanar with ABC. Intersection of p and \overline{AC} is the point B_1.
Find vector \overrightarrow{B_1B}.
Homework Equations
-Vector projection
- Dot...
Hello
First of all, I was told this is a physics problem. If it's incorrect, I apologize.
I have an image
I want to know the length of the black ship (the ship which is carrying a ship). Let's assume that I know the length of the text on the ship (I have a reference object).
Is this...
Please bear with me because I'm only in Pre-calculus and am taking basic high school physics. This is completely outside of my realm but curiosity has taken the better of me.
I just learned last week about the difference between Euclidean Geometry and Riemmanian Geometry (from another thread...
I am trying to build up a kind of mind map of the following:
Module (eg. vector space)
Ring (eg Field)
Linear algebra (concerning vectors and vector spaces, from what I understood)
Multilinear Algebra (analogously concerning tensors and multi-linear maps)
Linear maps & Multilinear maps
The...
I need help to visualize the geometry involved here,
How can I visualize the last paragraph? Why is the surface of fixed r now an ellipsoid? Also for r = 0, it is already a disk? I've tried searching for the geometry of these but I can't find any image of the geometry that I can just stare...
Homework Statement
A square with each side of 5 cm in length.Now if 4 parallel wires in each 4A current is flowing were placed on the vertex of the square.How can I find the center of the square of the magnetic strength?
Homework Equations
I am not sure what equation should be used.If I knew I...
I'd appreciate some explanation on how does one understand/reconcile the seemingly alternative concepts of gravity as (i) due to the warping of space by matter vs (ii) the exchange of gravitons. Is the latter a construction of how gravity can be considered within a quantum mechanics framework ...
Reading a somewhat long and argumentative thread here inspired the following unrelated question in my mind:
Where does a 2 dimensional flat Lorentzian geometry depart from Euclidean geometry as axiomatized by Euclid? I.e. Euclid's axioms (in modern language) can be taken to be:
We can...
As I understand it, a Cartesian coordinate map (a coordinate map for which the line element takes the simple form ##ds^{2}=(dx^{1})^{2}+ (dx^{2})^{2}+\cdots +(dx^{n})^{2}##, and for which the coordinate basis ##\lbrace\frac{\partial}{\partial x^{\mu}}\rbrace## is orthonormal) can only be...
Hello!
I started recently to use ANSYS Workbench 15 to solve a stiffness problem of a three parts assembly. I have a general knowledge about FEM. I create the geometry in the Design Modeler by inserting the assembly from Solidworks 2010.
It is a three parts assembly like a bicycle's wheel. Two...
Hi everyone,
I have recently installed ANSYS 14.5. But after opening workbench, while trying to open Fluent I am getting problems like "Unable to start the geometry editor." and "Unable to start the Meshing editor." I have attached the error screen shot. Please help me out in this. Thanks a lot.
Homework Statement
If two vertex of a square of the side AB is A(1,2) and B(2,4) find other two vertex C and D?
Homework Equations
1. y-y1 = m ( x-x1)
2. m=y1-y2 / x1-x2
3. m1*m2= -1
4. (x-x1) / (x1-x2) =(y-y1)/(y1-y2)
5. m=tanA
6. If ax+by+c=0 then its parallel line is ax+by+k=0
The Attempt...
Hi,
I am just about to finish working through the integration chapter of calculus on manifolds, and I am wondering whether it would be better to get spivaks first volume of differential geometry (or another book, recommendations?) and start on that, or to finish calculus on manifolds first...
Simple and basic question(maybe not). How are rotations performed in differential geometry ?
What does the rotation matrix look like in differential geometry? Let us assume we have orthogonal set of basis vectors initially.
I am looking to calculate the angle between two geodesics. Can this...
I am taking my bachelor in geometric quantization but I have no real experience in differential geometry ( a part of my project is to learn that). So I find myself in need of some good books that cover that the basics and a bit more in depth about symplectic manifolds.
If you have any...
On the one hand there are Differential Geometry, Algebraic Geometry
On the other there are Euclidean geometry, Hyperbolic geometry and elliptical geometry
On the other there are Affine geometry, projective geometry.
How do they all link up? Or are they all a bit different.
I'm trying to get an intuitive feel for Minkowski space in the context of Special Relativity. I should mention that I have not studied (but hope to) the mathematics of topology, manifolds, curved spaced etc., but I'm loosely familiar with some of the basic concepts.
I understand that spacetime...
Separate questions:
1. What is the mathematical formalism where one can transform between field and geometry or they both being emergence?
2. What is the mathematical formalism that can describe QFT but not using the concept of fields nor particles. What are they called and current attempts at...
How, if at all, would differential geometry differ between the opposite "sides" of the surface in question. Simplest example: suppose you look at vectors etc on the outside of a sphere as opposed to the inside. Or a flat plane? Wouldn't one of the coordinates be essentially a mirror while...
Homework Statement
Volume of tetrahedron T.ABC = V
Point P is on the middle of TA, Q is on the expansion of AB making AQ = 2AB
A shape is made through PQ which is parallel to BC so that it cuts the tetahedron into 2 pieces.
What is the volume of the biggest piece?
The Attempt at a Solution
I...