Geometric Definition and 813 Threads

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. P

    Geometric Random Variable probability problem

    Homework Statement X is a geometric random variable with p = 0.1. Find: ##a. F_X(5)## ##b. Pr(5 < X \leq 11)## ##c. Pr(X=7|5<X\leq11)## ##d. E(X|3<X\leq11)## ##e. E(X^2|3<X\leq11)## ##f. Var(X|3<X\leq11)##Homework EquationsThe Attempt at a Solution Can someone check my work and help me? a...
  2. M

    Differential Geometric Methods in Biology

    Dear members, Do you maybe know some good reference for the application of differential geometric methods in biology, especially evolutionary theory, game theory, population ecology. I kinda feel that the application of mathematical methods in biology in general is a field of increasing...
  3. FallenApple

    I Is Euclidean Space Inherently Geometric or Just a Vector Space?

    Or is it something separate that acts on a geometric space? So we know that the Euclidean space is a vector space. But is it geometric? I ask this because in group theory, the group elements are the operators acting on another set, but clearly we see that this doesn't mean that the group...
  4. L

    B Product of Negative Real Numbers: A Geometric Proof

    What is the Geometric Proof for the product of two negative real numbers being a positive real number?
  5. PsychonautQQ

    I Simplicial complex geometric realization 1-manifold

    Prop 5.11 from John M. Lee's "Introduction to Topological Manifolds":If K is a simplicial complex whose geometric realization is a 1-manifold, each vertex of K lies one exactly two edges. This proposition confuses me. If we look at the geometric realization of a simplex with two vertices, then...
  6. S

    I Outer product in geometric algebra

    Hello! I am reading so very introductory stuff on geometric algebra and at a point the author says that, as a rule for calculation geometric products, we have that ##e_{12..n}=e_1\wedge e_2 \wedge ...\wedge e_n = e_1e_2...e_n##, with ##e_i## the orthonormal basis of an n-dimensional space, and I...
  7. L

    MHB Converging Geometric Series with Negative Values?

    Hiya everyone, Alright ? I have a simple theoretical question. In a decreasing geometric series, is it true to say that the ratio q has to be 0<q<1, assuming that all members of the series are positive ? What if they weren't all positive ? Thank you in advance !
  8. Y

    MHB Geometric Series with Complex Numbers

    Hello all, Three consecutive elements of a geometric series are: m-3i, 8+i, n+17i where n and m are real numbers. I need to find n and m. I have tried using the conjugate in order to find (8+i)/(m-3i) and (n+17i)/(8+i), and was hopeful that at the end I will be able to compare the real and...
  9. M

    MHB Can the geometric and arithmetic means be applied to algebraic expressions?

    Given two positive numbers a and b, we define the geometric mean and the arithmetic mean as follows G. M. = sqrt{ab} A. M. = (a + b)/2 If a = 1 and b = 2, which is larger, G. M. or A. M. ? G. M. = sqrt{1•2} G. M. = sqrt{2} A. M. = (1 + 2)/2 A. M = 3/2 Conclusion: G. M. > A. M. Correct...
  10. G

    Explore Geometric Locus of Triangle Orthoprojections

    Homework Statement [/B] Given a general triangle ABC, find the geometric locus of points such that the three orthoprojection onto the sides of the triangle are aligned. Homework Equations Let's call A', B', and C' the orthoprojection of a given point M onto (AB) , (BC) , and (AC). M satisfies...
  11. M

    MHB Should NYC High Schools Ban Geometric Proofs?

    Most high schools in NYC have banned direct and indirect geometric proofs. Too many complaints from parents and schools in fear of a lawsuit decided to disregard proofs in geometry. Good idea? Bad idea?
  12. DaTario

    A Math Journal with geometric constructions

    Hi All, I would like to know which are the journals of mathematics that publish papers on geometric constructions. Most of the journals on geometry I have found tends to an analytic approach. Best wishes, DaTario
  13. G

    I Is the Proof of Geometric Progression in Probability Common Sense?

    My Statistics textbook does not prove this. The author think it is commons sense. I am not sure about this proof. Thank you.
  14. W

    Geometric interpretation of complex equation

    Homework Statement $$z^2 + z|z| + |z|^2=0$$ The locus of ##z## represents- a) Circle b) Ellipse c) Pair of Straight Lines d) None of these Homework Equations ##z\bar{z} = |z|^2## The Attempt at a Solution Let ##z = r(cosx + isinx)## Using this in the given equation ##r^2(cos2x + isin2x) +...
  15. F

    MHB Finding sum of infinite geometric series

    find the sum of this infinite geometric series: 1 - √2 + 2 - 2√2 + ... a.) .414 b.) -2.414 c.) series diverges d.) 2 I found that the common difference is 2, so I calculated this: S∞= -.414/-1 s∞= .414 So i got that the answer is A, but will you check this?
  16. C

    Pole of a function, as a geometric series

    Homework Statement Determine the order of the poles for the given function. f(z)=\frac{1}{1+e^z} Homework EquationsThe Attempt at a Solution I know if you set the denominator equal to zero you get z=ln(-1) But if you expand the function as a geometric series , 1-e^{z}+e^{2z}... I...
  17. CCR5

    Geometric Optics and Lens Power

    Homework Statement A farsighted boy has a near point at 2.3 m and requires eyeglasses to correct his vision. Corrective lenses are available in increments in power of 0.25 diopters. The eyeglasses should have lenses of the lowest power for which the near point is no further than 25 cm. The...
  18. E

    Values of x for which a geometric series converges

    Need help with a homework question! The question gives: The first three terms of a geometric sequence are sin(x), sin(2x) and 4sin(x)cos^2(x) for -π/2 < x < π/2. First I had to find the common ratio which is 2cos(x) Then the question asks to find the values of x for which the geometric series...
  19. M

    B Kepler's 3rd Law, geometric relationship?

    Hello, quick question here I am studying mathematical astronomy / the history of , and I have noted that by raising the average sidereal period of any planet in our solar system, to the power .666666, that you are left with the average distance of that planet from the Sun, in AU I was told...
  20. N

    Geometric average versus arithmatic average

    Homework Statement I have a range of numbers numbers n_i, each with a different weight w_i that sum up to 1. To keep things simple, let's take the case where we have three numbers with the following weights: n_i w_i ------------------------------...
  21. M

    MHB What is the geometric description of a set of vectors in $\mathbb{R}^2$?

    Hey! :o We have two vectors $\vec{u}, \vec{v}\in \mathbb{R}^2$. I want to describe geometrically the set of vectors $\vec{z}$, for which it holds that $$\vec{z}=\lambda{u}+(1-\lambda)\vec{v}$$ with $0\leq \lambda \leq 1$. Does this set describe all the points that are on the line that...
  22. N

    I A new vector-product for geometric algebra?

    I am investigating the mathematical properties of a vector-product. I am wondering if it might be old-hat in GA (which is new to me)? I am using the working-title "spin-product" for a vector-product that combines RANDOM rotation-only of a direction-vector [a unit 1-vector; say...
  23. M

    MHB What is a geometric interpretation of all these information?

    Hey! :o We have the tableau $\begin{pmatrix} \left.\begin{matrix} 1 & 0 & \alpha \\ 0 & 1 & \beta \\ 0 & 0 & 0 \end{matrix}\right|\begin{matrix} c\\ d\\ 0 \end{matrix} \end{pmatrix}$ Since there is a zero-row, we conclude that the column vectors are linearly dependent. The number...
  24. J

    Quantum Are there any good books on Geometric Phases in Quantum Mechanics?

    Hello! I would really appreciate it if somebody could recommend to me any books on geometric phases especially on Berry's phase. Thanks!
  25. jedishrfu

    I Article on Quantum Bootstrapping and Geometric Theory Space

    Here's an interesting article from Quanta magazine: https://www.quantamagazine.org/20170223-bootstrap-geometry-theory-space/ and some backstory: https://en.wikipedia.org/wiki/Bootstrap_model
  26. binbagsss

    Moments from characteristic function geometric distribution

    Homework Statement Hi, I have the probabilty density: ##p_{n}=(1-p)^{n}p , n=0,1,2... ## and I am asked to find the characteristic function: ##p(k)= <e^{ikn}> ## and then use this to determine the mean and variance of the distribution. Homework Equations [/B] I have the general expression...
  27. Milsomonk

    I Geometric Algebra formulation of Quantum Mechanics

    Hi all, I'm reading a paragraph from "Geometric Algebra for Physicists" - Chris Doran, Anthony Lasenby. I'm quite interested in applying GA to QM but I've got to a stage where I am not following part of the chapter and am wondering if someone can shed a little light for me. The part...
  28. binbagsss

    Geometric series algebra / exponential/ 2 summations

    Homework Statement I want to show that ## \sum\limits_{n=1}^{\infty} log (1-q^n) = -\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty} \frac{q^{n.m}}{m} ##, where ##q^{n}=e^{2\pi i n t} ## , ##t## [1] a complex number in the upper plane.Homework Equations Only that ## e^{x} =...
  29. J

    I Extremal condition in calculus of variations, geometric

    Hi folks, I am a bit confused with the extreme condition used in the calculus of variations: δ = 0 I don't understand this rule to find extreme solutions (maximum or minimum) If in normal differential calculus we have a function y = y(x) and represent it graphically, you see that at the...
  30. karush

    MHB Use the techniques of geometric series

    $\tiny{242.WS10.a}$ \begin{align*} &\textsf{use the techniques of geometric series} \\ &-\textsf {telescoping series, p-series, n-th term } \\ &-\textsf{divergence test, integral test, comparison test,} \\ &-\textsf{limit comparison test,ratio test, root test, } \\ &-\textsf {absolute...
  31. karush

    MHB 206.10.3.17 Evaluate the following geometric sum

    $\tiny{206.10.3.17}$ $\textsf{Evaluate the following geometric sum.}$ $$\displaystyle S_n=\frac{1}{2}+ \frac{1}{8}+\frac{1}{32}+\frac{1}{128}+\cdots + \frac{1}{8192}$$ $\textsf{This becomes}$ $$\displaystyle S_n=\sum_{n=1}^{\infty}\frac{1}{2^{2n-1}}=\frac{2}{3}$$ $\textsf{How is this morphed...
  32. karush

    MHB 242.10.3.27 using the geometric formula of a sum

    $\tiny{242.10.3.27}$ evaluate $$S_j=\sum_{j=1}^{\infty}3^{-3j}=$$ rewrite $$S_j=\sum_{j=1}^{\infty} 27^{j-1}$$ using the geometric formula $$\sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{1-r}, \left| r \right|<1$$ how do we get $a$ and $r$ to get the answer of $\frac{1}{26}$ ☕
  33. karush

    MHB 10.3.54 repeating decimal + geometric series

    $\tiny{206.10.3.54}$ $\text{Write the repeating decimal first as a geometric series} \\$ $\text{and then as fraction (a ratio of two intergers)} \\$ $\text{Write the repeating decimal as a geometric series} $ $6.94\overline{32}=6.94323232 \\$ $\displaystyle A.\ \ \...
  34. karush

    MHB Series using Geometric series argument

    $\displaystyle\text{if} \left| r \right|< 1 \text{ the geometric series } a+ar+ar^2+\cdots ar^{n-1}+\cdots \text{converges} $ $\displaystyle\text{to} \frac{a}{(1-r)}.$ $$\sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{(1-r)}, \ \ \left| r \right|< 1$$ $\text{if} \left| r \right|\ge 1 \text{, the series...
  35. S

    I Origin of geometric similarities between multipoles & AO's

    a textbook I'm reading has pointed out geometric similarities between atomic orbitals and multipoles. do these similarities originate from a mutual dependence on the spherical harmonics? if so, how does something like a dipole or a quadrupole depend on the Ylm's? Note that my I did my...
  36. Pouyan

    Differential equations and geometric series

    Homework Statement I Have a differential equation y'' -xy'-y=0 and I must solve it by means of a power series and find the general term. I actually solved the most of it but I have problem to decide it in term of a ∑ notation! Homework Equations y'' -xy'-y=0 The Attempt at a Solution I know...
  37. M

    MHB How Do You Solve This Geometric Progression Problem?

    If the second term is 6 and the 5th term of a geometric progression is 48.Find the first term and the common difference of it The sum of certain number of terms of the above progression from first term is 381.Find the number of terms of it. Any ideas on how to begin ?
  38. J

    I What's the geometric interpretation of the trace of a matrix

    Hello, I was just wondering if there is a geometric interpretation of the trace in the same way that the determinant is the volume of the vectors that make up a parallelepiped. Thanks!
  39. karush

    MHB Calculate the sum for the infinite geometric series

    Calculate the sum for the infinite geometric series $4+2+1+\frac{1}{2}+...$ all I know is the ratio is $\frac{1}{2}$ $\displaystyle\sum_{n}^{\infty}a{r}^{n}$ assume this is used
  40. M

    MHB From a sketch to the compass ; Geometric construction

    This is a rough sketch, (Happy) Now apart from constructing the triangle can you help me to located the point D & Obtain the location of point E on side AB such that ACDE is a trapezium ,only using an straight edge and a compass. (Crying)
  41. J

    Applied Books like J. Callahan's Advanced Calculus: A geometric view

    Hello, do you know of any books similar in style to Callahan's Advanced Calculus book(a book that explains the geometrical intuition behind the math)? This goes for any subject in mathematics(but especially for subjects like vector calculus, differential geometry, topology). Thanks in advance!
  42. P

    MHB Understanding Complex Geometric Sequences: A Revision Question

    need a hand with a revision question, I don't quite understand how to go about solving it question is attached below
  43. M

    MHB Show that angle AXC=angle ACB; geometric construction

    Looks like this question is going to make a long thread. :) This is what the problem states Using the pair of compass and a straight edge, Then, After That, Thereafter, Thereafter, The question states to mark the point where the tangent and AD produced as 'X'.Now can you help me...
  44. A

    A Geometric Quantum mechanics -- Worked examples?

    I recently found that formulation of quantum mechanics as a hamiltonian flow in a Kahler manifold, where there is a classical hamiltonian, hamilton equations, poisson brackets and etc. And while the mathematics in terms of differential geometry is all fine and good, I'm having problem finding...
  45. alexandria

    Arithmetic and Geometric Series

    Homework Statement Homework Equations no equations required 3. The Attempt at a Solution a) so for part c) i came up with two formula's for the tortoise series: the first formula (for the toroise series) is Sn = 20n This formula makes sense and agrees with part a). for example, if the...
  46. D

    Geometric Inequality: Prove √(2x)+√(2y)+√(2z)≤√(x+y)+√(y+z)+√(x+z)

    Homework Statement Let a,b and c be lengths of sides in a triangle, show that √(a+b-c)+√(a-b+c)+√(-a+b+c)≤√a+√b+√c The Attempt at a Solution With Ravi-transformation the expressions can be written as √(2x)+√(2y)+√(2z)≤√(x+y)+√(y+z)+√(x+z). Im stuck with this inequality. Can´t find a way to...
  47. malawi_glenn

    I Is Geometric Algebra inconsistent/circular?

    I am trying to learn Geometric Algebra from the textbook by Doran and Lasenby. They claim in chapter 4 that the geometric product ab between two vectors a and b is defined according to the axioms i) associativity: (ab)c = a(bc) = abc ii) distributive over addition: a(b+c) = ab+ac iii) The...
  48. W

    Geometric Similarity: Solving for Lp and Lm | Prototype and Model Lengths

    Homework Statement in this question , Lm / Lp = 1/ 6 ? or Lp / Lm = 1/6 ? Lp = length of prototype , Lm = Length of model Homework EquationsThe Attempt at a Solution i really have no idea... can someone help please?
  49. B

    I Geometric efficiency and CTDI(vol)

    Is there a relationship between the geometric efficiency of the scanner and computer tomography dose index of a scanner? I expect the relationship would be inverse but I wanted to check (i.e: if I half the geometric efficency I would double the CTDI(vol). Is this correct thanks
  50. P

    Proving Brewster's Angle Without Fresnel Equations?

    Homework Statement Background from previous parts of the question: A simple isotropic dielectric occupies the region x>0, with vacuum in region x<0. I've found the wave equations for the electric field Incident, reflected and transmitted to prove Snell's law (Sinθ/Sinθ = c/c' = √εr) and the law...
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