What is Geometrical: Definition and 143 Discussions

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.

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  1. Leureka

    I How does EM wave geometrical attenuation affect atomic absorption?

    Let's say we have a point source of an EM wave in a vacuum of total energy E, and an absorber atom at some distance from this source, whose first excited state is at the energy B, with B < or = E. The energy of the wave is constant as a whole, but at each point around the source the energy...
  2. Anish Joshi

    B Geometrical Optics: Explaining the Effects of Small Wavelengths

    Read this in my textbook:- The reason Geometrical optics works in case of formation of shadows, reflection and rarefaction is that the wavelength of light is much smaller compared to the reflecting/refracting surfaces as well as shadow causing objects that we use in day-to-day life. I...
  3. K

    I MOND from MacDowell-Mansouri geometrical formulation

    One poster is a strong promoter of Deur and Deur theory of how to get MOND out of GR via self-interaction and analogy to QCD. There is intense skepticism of Deur's approach, which has 0 citations other than the author. I saw this paper, High Energy Physics - Theory...
  4. tomceka

    Geometrical optics: using Snell's law, find the depth of the pool

    α=30°; l=0.5 m; n1=1; n2=1.33 α+β=90°, so β=90°-30°=60°. Using Snell's law: sinβ/sinγ = n2/n1 sinγ≈0.651 γ≈41°. β=γ+θ (vertical angles) θ=60°-41°=19° tan(θ+β)=l/h h=l/tan(θ+γ) h=0.5/(tan(19+41))≈0.289 m
  5. Marioweee

    What is the focal point of a lens in a geometrical optics problem?

    I have recently started with geometric optics and I do not quite understand what this problem asks of me. According to the statement, the focal point of the lens would be -25.5cm, right? That is, it is only a problem of concepts where it is not necessary to take into account the radii of the...
  6. S

    B Geometrical meaning of magnitude of vector product

    My notes says that the geometrical meaning of $$|\vec v \times \vec w | $$ is the perpendicular distance from point ##V## to line passing through ##O## and ##W## (all vectors are position vectors) $$|\vec v \times \vec w | = |\vec v| |\vec w| \sin \theta$$ From the picture, the perpendicular...
  7. S

    Calculation of orbital velocity -- Geometrical solution

    Hello, I would like to calculate the orbital velocity using the geometrical way of reasoning. But I have a hard time to understand and apply some basics into my calculations. The reasoning is pretty simple. After some time: dt ,the particle travels the distance: Vtot1 * dt = R*sinθ (see the...
  8. I

    Geometry Geometrical books (differential geometry, tensors, variational mech.)

    I am looking for math books that focus on geometrical interpretations. Sadly most of the modern books lack these interpretations and only consists out of theorems and proofs. It seems to me that most modern mathematicians are pure left-brain sequential thinkers that do not have a lot of...
  9. OlegKmechak

    Is there some geometrical interpretation of force from Newton's Laws?

    dP = F dt dE = F dr or if we introduce ds = (dt, dr) (dP, dE) = F ds And both dP and dE are constant in closed system. Some questions: - How does its implies on definition of Force? - Is there some clever geometrical interpretation of Force? - Why P and E seems almost interchengable?
  10. Frabjous

    Classical Geometrical Mechanics by Talman

    Does anyone have any experience with this book? https://www.amazon.com/gp/product/3527406832/?tag=pfamazon01-20
  11. S

    I Is it possible to calculate this geometrical relationship between circles?

    A large cirlcle with radius 50 m contains a smaller circle with radius 7.4 m that is tangent to its surface internally. Is it possible to calculate what number of the small circle the larger circle can contain iside it in which all are tangent to its surface ... but without using trig. Functions
  12. S

    MHB Prove Triangle Inequality: AB/MZ + AC/ME + BC/MD ≥ 2t/r

    Given a triangle ABC and a point M inside the triangle ,draw perpendiculars MZ,MD,ME at the sides AB,BC,AC respectively. Then prove:\frac{AB}{MZ}+\frac{AC}{ME}+\frac{BC}{MD}\geq\frac{2t}{r} Where t is half the perimeter of the triangle and r is the radius of the inscribed circle
  13. A

    A Are there conditions for the vanishing of geometrical phases in QM?

    Are there theorems for sufficient and necessary conditions for the vanishing of Berry and/or Wilzeck-Zee phases for a given quantum mechanical system?
  14. A

    A Crossing degeneracies and geometrical phases

    Assume all the usual things for the usual things for the geometric phases: A Hamiltonian that depend on external parameters, Adiabatic evolution, cyclic evolution in parameter space and all that If through the evolution in parameter space there is no energy level crossing, then a eigenvector of...
  15. A

    I Variation of geometrical quantities under infinitesimal deformation

    This question is about 2-d surfaces embedded inR3It's easy to find information on how the metric tensor changes when $$x_{\mu}\rightarrow x_{\mu}+\varepsilon\xi(x)$$ So, what about the variation of the second fundamental form, the Gauss and the mean curvature? how they change? I found some...
  16. W

    B Struggles with the geometrical analogy

    {Moderators note: thread split from https://www.physicsforums.com/threads/a-geometrical-view-of-time-dilation-and-the-twin-paradox-comments.842793 } I am a novice about relativity, but I found this convoluted and very difficult to follow. I think the lengths in the different coordinate systems...
  17. I

    Geometrical Optics - Light ray angles on a spherical mirror

    I can't see how the textbook produces the following relationships between angles: $$ \theta = \phi + \alpha \qquad (1)$$ $$ 2\theta = \alpha + \alpha ' \qquad (2)$$ My thinking is that the exterior angle theorem for triangles was used to create expression ##(1)##, but I am unsure as to how...
  18. Krushnaraj Pandya

    Stability of geometrical isomers

    Homework Statement Among the following, which should be the most stable compound? 1)Cis-cyclohexane-1,2-diol 2)Trans-cyclohexane-1,2-diol 3)Cis-cyclohexane-1,3-diol 4)Trans-cyclohexane-1,3-diolHomework Equations -- The Attempt at a Solution My thought process is-cis isomers with adjacent OH...
  19. Physiona

    I'm struggling with an Arithmetic Geometrical question.

    Homework Statement The angles in triangle ABC form an increasing arithmetic sequence. The ratio of angles A:B:C can be written in the form 185:370:555 respectively. You are told that the total area of the triangle is 9 Length BC is Given the area of...
  20. K

    B Any geometrical meaning of multiplication of quaternions?

    Let's just talk about unit quaternions. I know that $$\left(\cos{\frac{\theta}{2}}+v\sin{\frac{\theta}{2}}\right)\cdot p \cdot \left(\cos{\frac{\theta}{2}}-v\sin{\frac{\theta}{2}}\right)$$ where ##p## and ##v## are purely imaginary quaternions, gives another purely imaginary quaternion which...
  21. A

    MHB Complex number geometrical problem

    Show geometrically that if |z|=1 then, $Im[z/(z+1)^2]=0$ I am unsure how to begin this problem. I have sketched out |z|=1 but can't work out how to sketch the Imaginary part of the question.
  22. I like Serena

    MHB TikZ Challenge 1 - Geometrical Diagram - Votes

    Hey all, 2 weeks ago I created a challenge to create a geometrical diagram, like a triangle, that is somehow interesting or impressive. Now the moment of truth is here. Please everyone, give your vote! Voting will close in 2 weeks time. Let me recap the submissions.I like Serena...
  23. I like Serena

    MHB TikZ Challenge 1 - Geometrical Diagram

    Who can make the most impressive, interesting, or pretty TikZ picture? This first challenge is to create a geometrical diagram, like a triangle, that is somehow interesting or impressive. We might make it a very complicated figure, or an 'impossible' figure, or use pretty TikZ embellishments...
  24. B

    I Geometrical interpretation of gradient

    In 'Introduction to Electrodynamics' by Griffiths, in the section of explaining the Gradient operator, it is stated a theorem of partial derivatives is: $$ dT = (\delta T / \delta x) \delta x + (\delta T / \delta y) \delta y + (\delta T / \delta z) \delta z $$ Further he goes onto say: $$ dT =...
  25. S

    Physical meaning of geometrical proposition

    Is it true that one straight line goes through only 2 points?? If no then how ?? If yes then why??
  26. DaTario

    I Is there a geometrical derivation of e

    Hi All, Mr. James Grime from Numberphile channel has said () that the Euler´s number e has basically nothing to do with geometry. I would like to know if there is any derivation of e based on geometrical arguments. Best Regards, DaTario
  27. J

    Quantum Book on gauge transformations/symmetry & geometrical phases?

    Hello! I will be attending a course on condensed matter physics with emphasis on geometrical phases and I was wondering if the are any good books on gauge transformations, gauge symmetry and geometrical phases that you know of. Thanks in advance!
  28. N

    I Is there any way to get a geometrical description of QM

    This is maybe one of my greatest gripes with QM, I have never seen a geometrical description of it. What I mean by geometrical, is a description of the given object in the 3D world we live in, not a description in Hilbert Space, is such a description even possible in principle? I've been...
  29. M

    MHB How Can We Geometrically Describe Vectors Orthogonal to the Vector $(-4,1,-3)$?

    Hey! How could we describe geometrically the vectors that are orthogonal to the Vector $(-4,1,-3)$ ?
  30. CollinsArg

    B Trying to represent a written geometrical description

    I'm reading an old book about Thales (Greek geometry), and I can't understand what the next part means, and how to represent it graphically, could you help me? thanks: It begin stating that if you divide an equilateral traingle with a perpendicular from a vertex on the opposite side, it'll be...
  31. F

    A A question about coordinate distance & geometrical distance

    As I understand it, the notion of a distance between points on a manifold ##M## requires that the manifold be endowed with a metric ##g##. In the case of ordinary Euclidean space this is simply the trivial identity matrix, i.e. ##g_{\mu\nu}=\delta_{\mu\nu}##. In Euclidean space we also have that...
  32. Victor Alencar

    A Geometrical interpretation of Ricci and Riemann tensors?

    I do not get the conceptual difference between Riemann and Ricci tensors. It's obvious for me that Riemann have more information that Ricci, but what information? The Riemann tensor contains all the informations about your space. Riemann tensor appears when you compare the change of the sabe...
  33. A

    Geometrical optics, refraction

    Homework Statement A spherical surface of roc 10 cm separates 2 media x and y of refractive indices 3/2 and 4/3 respectively. Centre of the spherical surface lies in the denser medium. An object is placed i x medium. For image to be real, the object distance must be ---- A) >90 cm B) <90 cm...
  34. F

    Geometrical optics question.

    Homework Statement A container with a layer of water (n=1.33) of 5 cm thick is over a block of acrylic (n=1.5) of 3 cm thick. An observer watches (perpendicularly from above) the lowest surface of the acrylic. What distance does it (the bottom surface) seems to be from the top of the water...
  35. J

    Applied Book recommendations on geometrical methods for physicists

    Hello, I would like to obtain a book that has to do with geometrical methods/subjects for physicists. When i say geometrical methods/subjects i mean things like Topology, Differential Geometry etc. Thanks!
  36. n7imo

    I Geometrical Problem: What is the Value of DC?

    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...
  37. frostfat

    Finding a New GPS Coordinate Between Two Lines on a Sphere

    Hi, I have an interesting problem. I have three GPS coordinates, creating two lines across the surface of a sphere (assuming the Earth is spherical). I want to be able to create a new line (across the surface of a sphere) with a gradient that is in between the gradient of the two existing...
  38. A

    Intersection of Hyperboloid & 2-Plane=Ellipse

    I want to try and see the intersection between the hyperboloid and the 2-plane giving an ellipse. So far I have the following: I'm going to work with ##AdS_3## for simplicity which is the hyperboloid given by the surface (see eqn 10 in above notes for reason) ##X_0^2-X_1^2-X_2^2+X_3^2=L^2## If...
  39. Orodruin

    Insights A Geometrical View of Time Dilation and the Twin Paradox - Comments

    Orodruin submitted a new PF Insights post A Geometrical View of Time Dilation and the Twin Paradox Continue reading the Original PF Insights Post.
  40. Vinay080

    Motivation for geometrical representation of Complex numbers

    I am seeing in "slow motion" the development of vectorial system. I am reading the book "A History of Vector Analysis" (by Michael J.Crowe); it seems from my acquaintance that the vector concept came from the quaternions concept; and the quaternions concept came from the act of search for...
  41. A

    Geometrical Optics: Image Position with No Principal Axis

    What would be the position an image formed by a Lens or a mirror if the object is not kept on the principal axis?
  42. bcrowell

    SR as Geometrical Constructions

    This is just a random thought, may be totally wrong. Euclidean geometry was originally described as a constructive theory in which the axioms state the existence (and implied uniqueness) of certain geometrical figures. These constructions are the ones that can be done with two concrete tools: a...
  43. Titan97

    Geometrical meaning of Curl(Gradient(T))=0

    What is the geometrical meaning of ##\nabla\times\nabla T=0##? The gradient of T(x,y,z) gives the direction of maximum increase of T. The Curl gives information about how much T curls around a given point. So the equation says "gradient of T at a point P does not Curl around P. To know about...
  44. isnainidiah

    Geometrical Optics: Explaining Concepts with Fermat's Principle

    How do you use Fermat's principle of least time to explain various concepts in geometrical optics?
  45. Christian Grey

    Astrophysics uses wave optics or geometrical optics?

    I know Astrophysics uses concepts like relativity etc. But I want to know does it uses wave optics or geometrical optics? The phenomenon of light,wave optics(reflection,refraction,polarization,diffraction and interference) that we see everyday, is used in Astrophysics? Or does it uses...
  46. evinda

    MHB Euler's method - geometrical explanation

    Hello! (Wave) We take into consideration the following ODE: $\left\{\begin{matrix} y'=2t &, 0 \leq t \leq 1 \\ y(0)=0 & \end{matrix}\right.$ Its solution is $y(t)=t^2$. The following graph shows geometrically how Euler's method work. $$y^{n+1}=y^n+hf(t^n,y^n)\\y^{n+1}=y^n+h \cdot 2 \cdot...
  47. K

    Curve Matching Techniques for Rotated Curves in Geometric Analysis

    I need to perform geometry matching of curves (see http://www.tiikoni.com/tis/view/?id=c54d9b8 ). As it can be seen, the big problem is that curves might be rotated, though they have similar shape. Do I need to make curve fitting and look at the parameters of analytical models? But, I guess...
  48. D

    Application of advanced spectrometer in geometrical optics?

    We have an advanced spectrometer in our geometrical optics lab! I'm seeking for any experiment in geometrical optics to include it!
  49. F

    Geometrical Proof: Prove Intersection Point on Line CM

    Homework Statement Consider an triangle ABC with M as the middle point of the side AB. On the straight line through AB you put the angle ∠ ACM at A and the angle ∠ MCB at B. Now you have two new lines. The new lines should be on the same side of AB as C. Proof that the intersection point of the...
  50. LiHJ

    Geometrical properties of circle

    Homework Statement Dear Mentors/PF helpers, Please help me with part (ii), I couldn't find a way through this part. Homework Equations The Attempt at a Solution My solutions: (i) angle RPS = 65 deg (angles in the same segment are equal) (iii) angle PRS = 110 - 65 = 45 ( exterior...