What is Geometry: Definition and 998 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. I

    1st year high school geometry proof

    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...
  2. DaveC426913

    B Deceptively simple geometry question on SAT test

    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...
  3. Lotto

    What is a focal length of a thin convex lens?

    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...
  4. Mohmmad Maaitah

    Engineering Hard time finding the angle

    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:
  5. deuteron

    Why does the given conserved quantity mean the motion is on a cone?

    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?
  6. S

    I All possible QFTs from geometry?

    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...
  7. A

    A Geometry of a gauge potential

    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...
  8. resurgance2001

    How does Rotating an Ellipse change its equation?

  9. DaveC426913

    B How does one draw a logarithmic scale?

    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...
  10. wrobel

    B A triangle problem: Prove that |AC|+|AB|=|PR|+|PQ| given some constraints

    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...
  11. Mr X

    I Resource help for co-ordinate geometry

    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...
  12. S

    Run MCNP 5 input file of certain geometry for flux calculation

    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...
  13. Charles Link

    So many people are unable to do simple algebra

    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 ##...
  14. S

    Radii of stacked circles inside the graph of y = |x|^1.5

    (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...
  15. M

    Can ChatGPT do geometry?

    Gave following to bing chat and chatgpt, Both gave wrong answers. Any thoughts? ------------------------------ 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...
  16. L

    Significance of a Mobius strip?

    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?
  17. M

    Can AI handle geometry problems?

    Can AI or GPT-4 answer the following? ------------------------------- 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?
  18. S

    I Geometry of series terms of the Riemann Zeta Function

    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...
  19. Jiketz

    Geometry: prove that point M is touched by 4 circles

  20. Lotto

    Where to find molecular geometry for Psi4?

    I suppose there is a database for that, but I can't find it. Could anyone send here a link?
  21. A

    Geometry problem with arcs in a triangle

    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...
  22. M

    A Wave energy dissipated due to geometry

    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...
  23. barryj

    B What geometry theorem is used in this figure?

    What geometry theorem is used to be able to state that 8/4 = x/6 ??
  24. VladStoyanoff

    I Given a triangle ABC, on which side of the triangle does the point D lie?

    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...
  25. murshid_islam

    Any other books like David Acheson's "The Wonder Book of Geometry"?

    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?
  26. N

    I Can I Solve y in y^2=ln(1-y^2)?

    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...
  27. chwala

    Prove that ##AE=2BC## -Deductive Geometry

    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...
  28. chwala

    Find the length of PA - deductive Geometry

    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!
  29. BadgerBadger92

    Studying What should I learn first: Trigonometry or Geometry?

    I’m teaching myself algebra right now so I’m not at that point, but I was wondering when I finish algebra what should I study next? Trig or Geometry?
  30. W

    I Relationship between geometry and expansion rate

    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...
  31. Kinetic cyclic scissors

    Kinetic cyclic scissors

    Fun with tiled cyclic quadrilaterals
  32. K

    I Noncommutative geometry and asymptotic safe gravity and MOND

    I saw 2 recent papers on MOND are they promising [Submitted on 21 Jul 2022] Noncommutative geometry and MOND Peter K.F. Kuhfittig Comments: 5 pages, no figures Subjects...
  33. kirkr

    A Schwarzschild Geometry: Evaluating Proper Distance

    Schwarzschild Geometry-proper distance. From what I have studied when the Schwarzschild line element is evaluated at constant time and at a constant radius , proper distance becomes a Euclidean distance on the surface of a sphere. What I don't understand is how to evaluate the integral...
  34. S

    I Questions concerning the geometry of spacetime

    I have some questions about the space of the rectangle shown in the spacetime diagram. The red and blue lines are world lines of objects at rest with each other. 1) Does the rectangle have an area? (if no please go to question 3) 2) Is the rectangle a 2d Euclidean space? (if no please go to...
  35. S

    B What the result of null added to itself?

    Hey,... is that correct to say that "The null part added to itself will always remain the same as itself, namely null."" In arithmetic, 3 * 0 = 0 + 0 + 0 = 0; It is therefore healthy to extrapolate as follows: ∞-1 * 0 = 0 + 0 + 0 +... + 0 + 0 + 0 + 0 = 0 ∞ * 0 = 0 + 0 + 0 +... + 0 + 0 +...
  36. A

    I Energy reduction/deflection of beta particles due to isotope geometry

    Hello all. I'm an undergraduate student looking to conduct an experiment with an isotope that undergoes beta decay. I am curious as to the degree to which the isotope geometry will reduce the energy of/deflect beta particles emitted from the isotope. By geometry, I mean the "shape" of the...
  37. S

    MHB Euclidean Geometry - Demonstration Exercise

    (a) Let be m a line and the only two semiplans determined by m. (i) Show that: If are points that do not belong to such , so and are in opposite sides of m. (ii) In the same conditions of the last item, show: and . (iii) Determine the union result , carefully justifying your answer...
  38. A

    Vehicle cornering behavior and Ackermann geometry

    Dear all, I am writing a vehicle dynamics simulation for my thesis topic. However, I came into a conundrum when testing the cornering behavior of my vehicle. The problem is inherently complex due to its many subsystems, but I'll try to give as much detail without bogging the thread down...
  39. Trysse

    B: Calculate the distance between two points without using a coordinate system

    Dear all, the following problem is not a home-work problem. I have come up with this question for myself. Nevertheless, I am stuck and need your help. The question is: Can I calculate the distance between points A and B from this information? And if yes, how? I think it should be possible...
  40. jedishrfu

    B Flattening Object with Computational Geometry

  41. B

    B Distance between two points on Earth doesn't increase with altitude?

    Consider the following example: Point A has coordinates 45 lat, 0 long. Point B has coordinates 45 lat, 2 long. Both points are 5000 ft above sea level. The distance between them is X. Point C has coordinates 45 lat, 100 long. Point D has coordinates 45 lat, 102 long. Both points are at sea...
  42. theycallmevirgo

    Simple(?) parabolic dish design question

    I want to use this to design a parabolic (optical) mirror; The problem is that in my application I need both D and f to be a parameter, but I need to specify f only as a perpendicular distance from D. In other words, I need to specify some f_2=f-d, and calculate d. I can't seem to come up with...
  43. Perchaddition

    B Can I use the circle circumference formula for a sphere?

    Trying to calculate a circumference of a sphere from a radius of 3.09 inches. Is 19.4 a correct answer? Just ran numbers in the first circumference calculator I found http://calcurator.org/circumference-calculator/. Can I use the same formula for a sphere? What can I say ...Geometry is not my...
  44. Somaiyah

    Help with deciding electives: Differential Geometry or Quantum Info

    Hello everyone, I wanted some help deciding which elective to choose. I am a junior and for my next semester I have the option to pick either Differential Geometry-I or Quantum Information. I am confused which one to choose. We will be doing QMII as a compulsory course next semester and I have...
  45. qwertypo

    Why in this short circuit geometry can I not sense a photocurrent?

    Hi, I'm wondering why shorted circuit geometry like figure 2 did not sense photocurrent? Even if the the circuit composed like 2, I guess that by the Kirchhoff's Law, voltage should apply to the ampere meter and photocurrent should be sensed. But in real experiment, I found that shorted circuit...
  46. MidgetDwarf

    Geometry Any major difference between editions of Geometry David Brannan?

    Are there any major difference between the first and second edition of Geometry, by David Brannan. Besides price of course.
  47. S

    I Linear Algebra 1 problem, Vector Geometry: Lines

    Problem: Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1). I know the answer is (3 ± 2 / √5, -1 ± 4/√5) but I don't know where to start. I found that if t=2, x= (-5, 5) and the normal vector is (2, 1) but I am not sure if this information is useful or how...
  48. Twigg

    A Why are only TM waves allowed in this geometry? (Planar interface)

    I'm reading about excitation of surface plasmons, and there's a claim in the derivation I don't know how to prove. The geometry is two infinite slabs of material with negligible permeability (##\mu_1 = \mu_2 = 1##) and different permittivity ##(\epsilon_1 \neq \epsilon_2 \neq 1)##. The claim...
  49. chwala

    Solve this Geometry Problem using Alternate Segments

    This is a textbook problem...clearly angle ##AEG≠ 160^0## there is something wrong with the value given,...I am trying to analyse this...
  50. docnet

    Simple geometry problem -- Find the perimeter of the triangle ABC

    Hello, so I saw this problem on a website while looking up trigonometric identities and trying to solve it. what I know: The internal angles add up to pi Let the tangent point between A and B be X Let the tangent point between B and C be Y Let the tangent point between C and A be Z ##...