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. N

    Geometric Interpretation of VSEPR Theory

    I am making a Geometric model for VESPR theory, which states that valence electron pairs are mutually repulsive, and therefore adopt a position which minimizes this, which is the position at which they are farthest apart, still in their orbitals. For example, the 2 electron pairs on either side...
  2. E

    What is the geometric interpretation of the vector triple product?

    The interpretation of the vector product is the area of the parallelogram with sides made up of a and b and the scalar triple product is the volume of the parallelpiped with sides a, b, and c, but what is the interpretation of the vector triple product. Is it just simply the area of the...
  3. C

    Should optical cables be water tight? Geometric optics

    Homework Statement Explain the physical principle of total internal reflection used by optical cables. Calculate the critical angle of incidence that corresponds to a refracted angle θair = 90 Next, calculate the critical angle for a bare glass fiber submerged in water nH2O = 1.33...
  4. L

    Solving Digitoxin Rate of Elimination with Geometric Progression

    Homework Statement Patients with certain heart problems are often treated with digitoxin, a derivative of the digitalis plant. The rate at which a person's body eliminates digitoxin is proportional to the amount present. In 1 day, about 10% of any given amount of the drug will be eliminated...
  5. C

    Geometric optics - thickness of acrylic ?

    Homework Statement A ray is deflected by 2.37cm by a piece of acrylic. Find the thickness t of the acrylic if the incident angle is 50.5 degrees. http://imgur.com/kx2VT5c Homework Equations n1sinΘ1 = n2sinΘ2 The Attempt at a Solution n of acrylic is 1.5. Therefore, the refracted...
  6. MexChemE

    Geometric interpretation of partial derivatives

    Good afternoon guys! I have some doubts about partial derivatives. The other day, my analytic geometry professor told us that slopes do not exist in three-dimensional space. If that's the case, then what does a partial derivative represent? Given that the derivative of a function with respect to...
  7. jdawg

    Making a geometric series that converges to a number

    Homework Statement How do you come up with a geometric series that converges to a number like 2? I'm kind of confused on how to work backwards through the problem. If someone could provide me with an example, that would be great! Homework Equations The Attempt at a Solution
  8. S

    How to solve a geometric distribution problem with a biased coin?

    A boy is playing with a biased coin. The probabilty of obtaining a head with the coin is 0.4. Determine the probability that the boy will require at least eleven tosses before obtaining his third head. I have been trying but can't get it at all... Can someone please explain me how to solve...
  9. D

    Geometric Proof: Proving Independence of DE + DF in Isosceles Triangle ABC

    Homework Statement In triangle ABC, AB = AC, and D,E,F are points on the interiors of sides BC,AB,AC respectively, such that DE perpendicular to AB and DF perpendicular to AC. Prove that the value of DE + DF is independent of the location of D Homework Equations So far we have all the...
  10. M

    How Do You Calculate the Expected Value of Geometric Brownian Motion?

    Hi, I am trying to answer the following question: Consider a geometric Brownian motion S(t) with S(0) = S_0 and parameters μ and σ^2. Write down an approximation of S(t) in terms of a product of random variables. By taking the limit of the expectation of these compute the expectation of S(t)...
  11. S

    Geometric series involving logarithms

    Homework Statement A geometric series has first term and common ratio both equal to ##a##, where ##a>1## Given that the sum of the first 12 terms is 28 times the sum of the first 6 terms, find the exact value of a. Hence, evaluate log_{3}(\frac{3}{2} a^{2}+ a^{4}+...+ a^{58}) Giving...
  12. W

    Understanding Geometric Optics: The Role of Ray Intersection in Image Position

    Why is the position of an image the intersection of 2+ rays?
  13. C

    Geometric Sequence (Only 4 terms and their sums are given)

    Homework Statement "In a geometric sequence, the sum of t7 and t8 is 5832, the sum of t2 and t3 is 24. Find the common ratio and first term." Homework Equations d = t8/t7 or t3/t2 tn = a * rn-1 The Attempt at a Solution So I thought of developing a system of equations then solving...
  14. S

    How Do You Derive the Common Ratio in a Geometric Series?

    Homework Statement In a geometric series, the first term is ##a## and the last term is ##l##, If the sum of all these terms is ##S##, show that the common ratio of the series is ##\frac{S-a}{S-l}##Homework Equations Sum of geometric seriesThe Attempt at a Solution I was thinking to use the sum...
  15. B

    Alternative Bound on a Double Geometric Series

    If |a_{mn}x_0^my_0^n| \leq M then a double power series f(x,y) = \sum a_{mn} x^m y^n can be 'bounded' by a dominant function of the form \phi(x,y) = \tfrac{M}{(1-\tfrac{x}{x_0})(1-\tfrac{y}{y_0})}, obviously derived from a geometric series argument. This is useful when proving that analytic...
  16. MarkFL

    MHB CALCULUS: Find Integral from -8 to -2 by Interpreting in Terms of Areas

    Here is the question: I have posted a link there to this thread so the OP can see my work.
  17. B

    Word Problem with Geometric Series

    Homework Statement The total reserves of a nonrenewable resource are 600 million tons. Annual consumption, currently 20 million tons per year, is expected to rise by 1% each year. After how many years will the reserve be exhausted? Part 2. Instead of Increasing by 1% each year, suppose...
  18. P

    Where Does the 0.61 in the Diffraction Limit Formula Come From?

    An infinity corrected microscope objective has a magnification of 100× for a tube lens with focal length 180 mm. The numerical aperture of the objective is 0.90. Calculate the the diffraction limited spatial resolution if the objective is used with red light (660 nm). (Ans.: f=1.8 mm; d=447...
  19. J

    Proper form for geometric series with a neg inside

    Homework Statement Curious about this ...I have to find the sum. Homework Equations The Attempt at a Solution Ʃ (1/4)(-1/3)^n from 1 to infinity I want to know the proper form and why. Is it (1/4) Ʃ (-1/3)(1/3)^n-1 or (1/4) Ʃ (1/3)(-1/3)^(n-1) You get different answers
  20. JJBladester

    Geometric Mean vs. Arithmetic Mean in Bandpass Filters

    Why is the geometric mean used to define the center frequency of a bandpass filter instead of the arithmetic mean? I read in this book that 1. All the lowpass elements yield LC pairs that resonate at ω = 1. 2. Any point of the lowpass response is transformed into a pair of points of the...
  21. F

    Eigenspaces and geometric reasoning

    Homework Statement Let T be the reflection about the line 6x + 1y = 0 in the euclidean plane. Find the standard matrix A of T. Then, write down one of the eigenvalues and its corresponding eigenspace (in the form span {[ ]}). Then, find the other eigenvalue of A and its corresponding...
  22. N

    MHB Why is my solution for part b) of the geometric series question incorrect?

    Please refer to the attached sheet. I need help with part b) for part a) I did: $\sum\limits_{n=0}^{\infty} a^n = \frac{1}{1-a}$ So for $\sum\limits_{x=0}^{\infty} a^{2x}$ $a^{2x} = (a^2)^x$ and $\sum\limits_{x=0}^{\infty} (a^2)^x = \frac{1}{1-a^2}$ for part b) the solutions say i am wrong...
  23. N

    MHB Geometric rv and exponential rv question

    Please refer to the attached image.for part a) this is what i did: $G = k$, $k-1< X < k$ so I substituted $k-1$ and $k$ into the given exponential rv, this gave me $\lambda e^{-\lambda(k-1)}$ and $\lambda e^{-\lambda k}$ $= \lambda e^{-\lambda(k-1)} + \lambda e^{-\lambda k}$ But I...
  24. L

    Geometric phase of a parallel transport over the surface of a sphere

    I have this question on the calculation of the geometric phase (Berry phase) of a parallel transporting vector over the surface of a sphere, illustrated by Prof. Berry for example in the attached file starting on page 2. The vector performing parallel transport is defined as ψ=(e+ie')/√2...
  25. P

    Interesting application of geometric algebra

    For those unaware, geometric algebra is a mathematical language that generalizes and simplifies a lot of the tools physicists work with (vectors, complex numbers, tensors, etc). I found a neat example in classical mechanics to illustrate its power. Simply, a charge in a constant magnetic field...
  26. K

    MHB Geometric action of an arbitrary orthogonal 3x3 matrix with determinant -1

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  27. marcus

    A purely geometric path integral for gravity

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  28. P

    Root 2 irrationality proof (geometric)

    I was looking over this proof and I have some questions: http://jeremykun.com/2011/08/14/the-square-root-of-2-is-irrational-geometric-proof/ Second paragraph, what does "swinging a b-leg to the hypotunese" mean? Also, where did the arc come from, I really don't understand also, the last part...
  29. P

    How Is the Angle Determined for the Opposite String in a Suspended Rod Scenario?

    A uniform rod of 80 Newtons is suspended from the ceiling by strings attached to its ends. The rod is in equilibrium at an angle of 10 degrees to the horizontal, and the string attached to the higher end is at an angle of 40 degrees tohe vertical. Find the angle which the other string makes with...
  30. Q

    Geometric Dimensioning and Tolerancing

    Can someone recommend a great textbook or resource for geometric dimensioning and tolerancing that would be appropriate for self-learners? An introductory text would be good, but better would be a textbook that covers it in depth.
  31. sas3

    New geometric version of quantum field theory

    The new geometric version of quantum field theory could also facilitate the search for a theory of quantum gravity that would seamlessly connect the large- and small-scale pictures of the universe https://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics/ I...
  32. L

    Can the dot and cross product prove the sum of squares in a parallelogram?

    Homework Statement Using vectors, the dot product, and the cross product, prove that the sum of the squares of the diagonals of a parallelogram is equal to twice the sum of the squares of two adjacent sides of the parallelogram. Homework Equations |A·B|=|A||B|cosθ |AxB|=|A||B|sinθ The...
  33. H

    MHB Geometric Puzzle: Find P & Q in R2 Square

    Find two sets P and Q satisfying I) P and Q are completely contained in the square, in R2, with vertices (1, 1), (1, -1), (-1, -1), and (-1, 1). II) P contains the ponts (1, 1) and (-1, -1) while Q contains (-1, 1) and (1, -1). III) P and Q are disjoint. IV) P and Q are both connected sets.
  34. B

    Proofing A Geometric Statement

    I've attached a picture that pertains to the query that I have. I have been able to show that each angle B is 60 degrees, but I am unsure as to how to show a is also a 60 degree angle. Isn't there some geometric theorem I could use?
  35. Saitama

    Geometric progression related problem

    Homework Statement The sum of the squares of three distinct real numbers, which are in geometric progression is ##S^2##. If their sum is ##\alpha S##, show that ##\alpha^2 \in (1/3,1) \cup (1,3)##. Homework Equations The Attempt at a Solution Let the three numbers be ##a , \, ar...
  36. E

    Geometric expressions for a quarter circle cut at an arbitrary point

    Homework Statement I am after finding general geometric expressions for a quarter-circle that is split into two segments along either its domain or range (they are equal). I.e. Taking the circle shown in Figure 1 and concentrating on the upper right quadrant, I am after expressions for the...
  37. B

    Is the geometric multiplicity of an eigenvalue a similar invariant?

    If two matrices similar to one another are diagonalizable, then certainly this is the case, since the algebraic multiplicity of any eigenvalue they share must be equal (since they are similar), and since they are diagonalizable, those algebraic multiplicities must equal the geometric...
  38. C

    Proving Congruence of Geometric Figures with Superposition

    Show that if a geometric figure is congruent to another geometric figure, which is in its turn congruent to a third geomtric figure, then the first geometric figure is congruent to the third. Answer : I will be showing what the question asks by using superposition of the geometric figures...
  39. rsyed5

    MHB How to Solve a Geometric Sequence with Given Differences?

    I have no idea how to solve this equation, its in my homework... i know the formula to find the nth term(tn=ar^n-1) but don't know how to solve this: The difference between the first term and second term in a geometric sequence is 6.The difference between the second term and the third term is...
  40. Fernando Revilla

    MHB Find x for Geometric Progression: Solve with Step-by-Step Explanation

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  41. B

    A geometric sequence within a arithmetic sequence

    the main question here is that can a sequence * arithmetic * be correct if the difference is also changing in terms of a geometric sequence ?\ now look at this sequence 0.33,0.3333,0.333333 now if we calculate the difference between the first two terms its 0.0033 between the second and...
  42. P

    Hyperbola Fermat, Geometric Infinite Sum

    Hello everybody, I'm trying to understand some steps in the evolution of calculus, and in a .pdf found in the internet I read the document: http://www.ugr.es/~mmartins/old_web/Docencia/Old/Docencia-Matematicas/Historia_de_la_matematica/clase_3-web.pdf , in pags. 14-15. I want to solve the to...
  43. B

    When is algebraic multiplicity = geometric multiplicity?

    In my last Linear Algebra class we saw Eigenvalues and Diagonalizations. It turns out that an n x n matrix is diagonalizable if its eigenbasis has n linearly independent vectors. If the characteristic equation for the matrix is (λ - λ_1)^{e_1}(λ - λ_2)^{e_2}...(λ - λ_k)^{e_k} = 0 then 1)...
  44. B

    Non-Integrability of a Pfaffian - Geometric Interpretation?

    The question of Solving a Pfaffian ODE can be interpreted as the question of finding the family of surfaces U = c perpendicular to a surface f generated by the vector field $$F(x,y,z) = (P(x,y,z),Q(x,y,z),R(x,y,z))$$ At each point, the gradient of the family of surfaces U = c will either...
  45. J

    MHB Geometric Sequences help - (3 given terms, find the rest)

    I need to find the value of the first term for this geometric series. Sn = 33 tn = 48 r = -2 I know that I have to take the formulas tn = t1 x r^(n-1), and Sn = [t1 x (r^n) - 1] / (r - 1), and isolate t1 for the first formula and then input that into the second, but I don't know the actual...
  46. dkotschessaa

    Geometric understanding of Semi-direct product

    In undergraduate abstract algebra we are not exposed to semi-direct products, so I was hoping someone could help me as I am doing some research in this area. I am familiar with the definitions of direct products and normal groups, and I know that a semidirect product is one where one of the...
  47. H

    Arithmetic progression used to determine geometric progression

    Homework Statement an arithmetic progression(a1-a9) has 9 numbers. a1 equals 1 The combination(S) of all of the numbers of the arithmetic progression is 369 a geometric progression(b1-b9) also has 9 numbers. b1 equals a1(1) b9 equals a9(unknown) find b7 Homework Equations...
  48. Lebombo

    Is this sequence arithmetic, geometric, or neither?

    Homework Statement Is the sequence \frac{1}{1}, \frac{1}{2}, \frac{1}{3} , \frac{1}{4}...\frac{1}{n} arithmetic or geometric? Homework Equations Common difference and Common ratio formulas The Attempt at a Solution I found the common difference from a_{2} - a_{1} =d_{1} and common...
  49. Lebombo

    Difference between 2 Sum of n terms of geometric series formulas

    Difference between 2 "Sum of n terms of geometric series" formulas Notation A) S_{n}= \sum_{k=0}^{n - 1} ar^{k} = ar^{0} + ar^{1} + ar^{2} +...+ ar^{n-1} = \frac{a(1-r^{n})}{1-r} Proof: S_{n}= ar^{0} + ar^{1} + ar^{2} +...+ ar^{n-1} - r*S_{n}= ar^{1} + ar^{2} + ar^{3} +...+ ar^{n}...
  50. Government$

    Arithemtic and geometric progession

    Homework Statement Numbers a,b,c are consecutive members of increasing arithmetic progression, and numbers a,b,c+1 are consecutive members of geometric progression. If a+b+c=18 then a^2 +b^2 + c^2=?The Attempt at a Solution a + b + c= 18 a + a +d +a + 2d = 18 3a + 3d = 18 3(a+d)= 18 a+d=6=b...
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