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. G

    Prove PGL(V) Acts 2-Transitively on P(V) Projective Space

    Dim(V)>1.Prove that PGL(V) acts two transitively but not 3 transitively on P(V) projective space.
  2. N

    Greens theorem and geometric form

    <y-ln(x^2+y^2),2arctan(y/x)> region : (x-2)^2+(y-3)^2=1 counter clockwise taking int int dQ/dx - dP/dy dA leads to -int int dA here my text is showing the next step as a solution of -pi not sure ..polar cords ext..
  3. A

    Simplifying a geometric series with an infinity summation bound

    Homework Statement I am solving some convolutions, and i have come to these solutions. a)\sum2k, summing from -\infty to -1 b)\sum2k, summing from -\infty to n , where n <=-1Homework Equations the geometric series summation formula, from 0 to N \sumak = 1-aN+1 / 1-a , summing from 0 to N The...
  4. T

    Arithmetic and geometric progression

    If the fourth, seventh and sixteenth terms of an AP are in geometric progression, the first six terms of the AP have a sum of 12, find the common difference of the AP and the common ratio of the GP. I've been assuming that the fourth, seventh and sixteenth terms of the AP are the fourth...
  5. srfriggen

    Linear Transformation; Geometric Representation

    Homework Statement (note; all column vectors will be represented as transposed row vectors, and matrices will be look like that on a Ti-83 or similar) L: R^3 -> R^2 is given by, L([x1, x2, x3]) = [2x1 + x2 - x3 x1 + 3x2 +2x3]* *Matrix Relevant...
  6. T

    Finding Common Ratio of Geometric Progression for 3 Points on a Parabola

    Homework Statement (p,a) , (q,b) and (r,c) are the coordinates of three points on the parabola y^2=3x. If the x-coordinate for these three points form a geometric progression whereas the corresponding y-coordinate form an arithmetic progression, find the common ratio of the geometric...
  7. N

    MATLAB code to Geometric Random Variable

    Homework Statement Generate Geometric RV with Porbabilty of succcess 0.1 using only rand() Homework Equations rand() geometric rv P=(1-p)^(k-1) * p where p=0.1, k is number of trial in which we get 1st success The Attempt at a Solution rand(n)
  8. F

    Geometric interpretation of the spacetime invariant

    For a euclidean space, the interval between 2 events (one at the origin) is defined by the equation: L^2=x^2 + y^2 The graph of this equation is a circle for which all points on the circle are separated by the distance L from the origin. For space-time, the interval between 2 events is...
  9. T

    What is the Geometric Progression for Rabbit Population Growth?

    Homework Statement Number of rabbits reared by Alice at the beginning of certain year is given as b. End of that particular year, the number of rabbits were given as 10+(3/2) b . Write down the number of rabbits at the end of second and third year. Find the total number of rabbits at the end...
  10. M

    Show that the inequality is true | Geometric Mean

    Homework Statement Let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Show that the inequality (1+R_{G})^{n} \leq V is true. Where R_{G} = (r_{1}r_{2}...r_{n})^{1/n} and V= \Pi_{k=1}^{n} (1+r_{k}) Homework Equations The Attempt at a Solution I've...
  11. M

    Prove this inequality : Geometric Mean and Arithmetic Mean

    Homework Statement let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Suppose an investment of one dollar at the beginning of the year k grows to 1+r_{k} at the end of year k (so that r_{k} is the "return on investment" in year k). Then the value of an investment of one dollar at...
  12. M

    Infinite geometric series problem

    Homework Statement \sum_{n=1}^\infty \frac{(-3)^{n-1}}{4^n} The Attempt at a Solution \sum_{n=1}^\infty \frac{(-3)^n-1}{4^n} \frac{1}{4}\sum_{n=1}^\infty \frac(-{3}{4})^{n-1} Can some one please explain how they got from the first step to the 2nd. How do you pull...
  13. jegues

    Taylor Series using Geometric Series and Power Series

    Homework Statement See figure attached. Homework Equations The Attempt at a Solution Okay I think I handled the lnx portion of the function okay(see other figure attached), but I'm having from troubles with the, \frac{1}{x^{2}} \int x^{-2} = \frac{-1}{x} + C How do I...
  14. jegues

    Exploiting Geometric Series with Power Series for Taylors Series

    I'm confused between some formulae so I'm going to give some examples and you can let me know if what I'm writing is correct. Find the Taylor series for... EXAMPLE 1: f(x) = \frac{1}{1- (x)} around x = 2 Then, \frac{1}{1-(x)} = \frac{1}{3-(x+2)} = \frac{1}{3} \left( \frac{1}{1...
  15. M

    Geometric/Berry Phase Explained | Elementary References

    Hi could someone please explain what Geometric/berry phase is I've had a look and there seems to be several ways to interpret the physics. My understanding is that it occurs when your quantum state traces out a closed path in some parameter space, which is some how related to degeneracies in a...
  16. N

    Find the first three terms a geometric sequence

    Homework Statement Find the first three terms of a geometric sequence given that the sum of the first four terms is 65/3 and the sum to infinity is 27. Homework Equations \begin{array}{1} S_n = \frac{a(1 - r^n)}{1 - r}\\ S_n = \frac{a(r^n - 1}{r - 1} \end{array} The Attempt...
  17. N

    Finding x in a geometric progression, given the sum.

    Homework Statement If 1 + 2x + 4x^2 + ... = \frac{3}{4} find the value of x. [Edit: Forgot to ask the question] Homework Equations S_n = \frac{a(1 - r^n)}{1 - r} t_n = ar^{n-1} The Attempt at a Solution a = 1 r = 2x I try to solve S_n and end up with 2x^n = \frac{6x - 7}{4}...
  18. K

    Geometric probability question

    A dog is running around in a fenced-off rectangular field with dimensions of 40ft by 50ft. If the position of the dog is uniformly random throughout the field, what is the probability that the dog is 10 feet or more away from the fence at any given time? In know that since the dog is...
  19. N

    Geometric series. Find the sum of the series. Powers.

    Homework Statement Find the sum of 9 terms of the series 3 + 3^(4/3) + 3^(5/3) + ... Homework Equations I'm just learning sequences and series and senior high school level. I'm finding it hard to apply a, ar, ar^(n-1), ... to this. a = 3. I don't know how to find common...
  20. K

    Find k for Geometric Sequence b1=1000, bn=(2/3)bn-1: 0.001

    b1,b2,b3,... In the geometric sequence above, b1=1000 and bn=(2/3)bn-1 for all n\geq2. What is the least value of k for which bk<0.001? The Attempt at a Solution What I did first was I found what b0 is since we are given b1 and that is 1500. But I do not understand where the k is...
  21. R

    IMPORTANT - what is the geometric intepretation of the gradient vector?

    IMPORTANT! ---- what is the geometric intepretation of the gradient vector? Assume the situation in which I have a slope, a component of a function dependent on x and y, which is at an angle to the xy plane. The gradient vector would be perpendicular to the tangent plane at the point in which i...
  22. K

    Geometric optics and photography

    Hi there, I have a question about photography. We know that in geometric optics, a bunch of parallel rays which going into the len will focus on the focus (a point). But as we see, the image is a set of points on a 2 dimensional plane. It is quite confusing that a focus is only a point...
  23. T

    How Does the Minkowski Metric Explain Special Relativistic Effects?

    I'm learning about special relativity in its differential geometry formulation. I don't understand how special relativistic effects can be derived from the Minkowski metric. It isn't obvious to me where relative velocity comes in, or why this makes things look different. Can somebody explain how...
  24. Rasalhague

    Coexistence of Tensor and Geometric Products in Multilinear Algebra

    They aren't equivalent in general, but do they ever coincide, and, if so, under what conditions? I've seen both denoted by juxtaposition. Is there a way to tell, in such cases, which is meant, or is it necessary to always use a different notation for the tensor product when the geometric product...
  25. E

    Q4 - Arithmetic and Geometric series

    Homework Statement Let a1, a2, a3 denote the first three terms of a geometrical sequence, for which a1 + a2 + a3 = 26. a1 + 3, a2 + 4, a3 - 3 are the first three terms of an arithmetical sequence. Find the first term and the common quotient (ratio) of the geometrical sequence...
  26. Sirsh

    Geometric Proof: Finding Angles in an Isosceles Triangle

    Hello all, the picture i have attached is the question. http://img842.imageshack.us/img842/8921/geometricproof.png I've concluded that there are two i isosceles triangles in this one triangle. \anglePSQ + \angleQSR = 90degrees Finding the angle in one of the isosceles triangles...
  27. J

    How Do You Solve a Geometric Progression with Sum and Term Constraints?

    An infinite geometric progression has a finite sum. Given that the sum of the first two terms is 9 and the third term is 12. 1/ Find the value of the first term and the common ration r.
  28. Helios

    Geometric Series: Finding the nth Term

    This looks almost like a geometric series; 1, 2, 5, 14, 41, 122, 365, ... but each term is one less than three times the preceeding one. So is this a sequence or a series? What is a formula for the value of the nth term in terms of n?
  29. C

    Solve Geometric Optics: Convex & Concave Lenses, Focal Lengths

    Homework Statement Given a convex lens of focal length of (x+5) cm and a concave lens of focal length x cm. The 2 lenses are placed 30 cm apart coaxially i.e along the same axis with the convex lens on the left while the concave lens is on the right. A light bulb is placed to the left of the...
  30. W

    Finding the Sum of Geometric Sequences with Given Constraints

    Homework Statement The sum of the first six terms in a geometric sequence of real numbers is 252. Find the sum of the first four terms when the sum of the first two terms is 12. Homework Equations Sn = A1 - A1Rn divided by 1 - R R \neq 1 (I can't figured out how to make the...
  31. Saladsamurai

    Derivative of Geometric Series

    Homework Statement I am having trouble following what is going on in this solution. We are looking to find the expectation value of: f(x,y)=\frac{1}{4^{x+y}}\cdot\frac{9}{16} I have gotten it down to: E(X) = \frac{3}{4}\sum_{x=0}^\infty x\cdot\left(\frac{1}{4}\right)^x\qquad(1) We know...
  32. J

    Number of Terms in Geometric Progression 0.03 to ar^n-1

    find the number of terms in 0.03+0.06+0.12+...+ar^n-1
  33. F

    Geometric Distribution problem

    Question: If Y has a geometric distribution with success probability .3, what is the largest value, y0, such that P(Y > y0) ≥ .1? Attempt: So i represented the probability of the random variable as a summation Sum from y0= y0+1 to infinity q^(yo+1)-1 p ≥ .1 using a change of variables...
  34. F

    Geometric Q: Determine Image Position in Long Glass Rod w/ Refractive Index

    the questions is The left end of a long glass rod in diameter has a convex hemispherical surface in radius. The refractive index of the glass is . Distances are measured from the vertex of the hemispherical surface (to the right is positive for image distances). Determine the position of...
  35. S

    Geometric understanding of integration / surface area of sphere

    Hi everyone, I've browsed around the forum a bit and found that others have had the same problem as me, however, none of the answers help me a lot, so I thought to post a more specific question, I hope you don't mind. I'm having a problem with the surface area of a sphere, probably because...
  36. Q

    Is Geometric Quantum Mechanics a Viable Alternative to Hilbert Space?

    I ran across a paper today that I found rather interesting. The idea is that "there exists a geometry description other than the conventional description in a Hilbert space...". The gist of the paper is that the quantum phase space can be viewed as a complex projective space if the dimensions of...
  37. R

    Evaluating an Infinite Series (non geometric)

    Homework Statement http://bit.ly/9N9iLZ Evaluate: lim n-> infinity of Sum (from k = 1 to n) of sqrt(k/n) * 1/n Homework Equations taylor series? The Attempt at a Solution the above = lim n->infinity of Sum (from k = 1 to n) of k^1/2 / n^3/2 k approaches n so n^1/2 / n^3/2 ->...
  38. E

    Geometric Constructible Numbers

    Geometric Constructible Numbers... Hi, everyone. I have a question about geometric constructible numbers. I know that "if 'a' is constructible then [Q(a):Q]=2^n." But I heard that its inverse is not true. I want some counter examples about the inverse statement. (I have checked by googling 'i'...
  39. I

    What Transformations Convert e^t to a Damped Exponential Function?

    This is just to see if I remember? Please confirm, correct any errors, and answer the questions (q's in bold) Homework Statement What geometric transformations will "transform" f(t) = et \stackrel{transformations}{\rightarrow} \frac{mg}{b} * (1 - e-bt/m)? Homework Equations f(t) =...
  40. B

    Evaluating Infinite Geometric Series: a sub n (0.1)^n

    Homework Statement Let an (read 'a sub n') be the nth digit after the decimal point in 2pi+2e. Evaluate SUM (n=1 to inf) an(.1)^n (here, again, an is meant to be 'a sub n') Homework Equations As far as I can see, this is a partial sum of a geometric series. To find the nth...
  41. M

    I have two question about Geometric Progressions

    Homework Statement 1. Evaluate 4^1/3 . 4^-1/9 . 4^1/27 2. express 0.85555 ... as a farction . ( hint: write 0.85555= 0.8+0.05(1+0.1+0.01+...)) The Attempt at a Solution 1. well in this question i think the " r " is in the power ,,, and it's -1/3 but how to complete it ,,, what...
  42. B

    Geometric description of (simplicial)homologous cycles

    Hi, everyone: I am trying to understand the geometric interpretation of two simplicial cycles being homologous to each other. Let C_k(X) be the k-th chain group in the simplicial complex X, and let c_k be a chain in C_k(X) The algebraic definition is clear: two...
  43. S

    Isomers: Geometric and Diastereometric Isomers

    What is the difference between Diastereomers and Geometric Stereomers? Im a little confused on this one as I thought they were the same thing? Stevo1925
  44. H

    Solving a Geometric Problem with Friends

    A couple friends worked on this problem (for a week now...) Trying to show that a conformal bijective map that sends vertices of one rectangle to vertices of another rectangle on the complex plane has to be linear. I would appreciate any help, Thank you.
  45. A

    The Total Vertical Distance of a Ball Dropping from 10 Feet

    a ball is dropped from a height of 10 feet, each bounce is 3/4 of the height of the bounce before a)find an expression for the height hn to which the ball rises after it hits the floor for the nth time so hn= 10(3/4)n b) find an expression for the vertical distance Di the ball has...
  46. M

    Solving Geometric Series: 2*(-1/4)^(n-1)

    {sigma} 2*(-1/4)^(n-1) Could i treat this as a geometric series? i know geometric is in the form of ar^n but the n is (n-1) my A=2 my r= -1/4
  47. icystrike

    Geometric interpretation and vector problem

    Homework Statement They were asking for the geometric interpretation and the says its triangular prism with infinite right angles. I don't understand what they mean by that. Homework Equations The Attempt at a Solution
  48. W

    Similar Matrices & Geometric Multiplicity

    Homework Statement Prove that if two matrices are similar then they have the same eigenvalues with the same algebraic and geometric multiplicity. Homework Equations Matrices A,B are similar if A = C\breve{}BC for some invertible C (and C inverse is denoted C\breve{} because I tried for a...
  49. R

    Contraction map of geometric mean

    I have the following mapping (generalized geometric mean): y(i)=exp\left[{\sum_j p(j|i)\log x(j)}\right]\\ ,\ i,j=1..N where p(j|i) is a normalized conditional probability. my question is - is this a contraction mapping? in other words, does the following equation have a unique...
  50. R

    Summation of geometric number of iid exponentially distributed random variables

    Hello, I am having difficulty approaching this problem: Assume that K, Z_1, Z_2, ... are independent. Let K be geometrically distributed with parameter success = p, failure = q. P(K = k) = q^(k-1) * p , k >= 1 Let Z_1, Z_2, ... be iid exponentially distributed random variables with...
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