What is Geometric: Definition and 807 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. A

    A Another great mathematical problem: Quadrisection of a disc

    Along with the problem of squaring a circle and trisection of an angle, there is one more great problem: quarisection of a disc. You have a disk and have to dissect it into four parts of equal area with three chords coming from the same point on the disc's boundary (one of these chords is a...
  2. chengmo

    MCNP geometric coincidence

    How do we solve the geometric coincidence problem? I need a semi-cylinder that fits into the cuboid but if I use the cuboid and the cylinder directly it's geometrically problematic
  3. K

    I Geometric Distribution Problem Clarification

    (Geometric). The probability of being seriously injured in a car crash in an unspecified location is about .1% per hour. A driver is required to traverse this area for 1200 hours in the course of a year. What is the probability that the driver will be seriously injured during the course of the...
  4. dom_quixote

    B Geometric Issues with a line, a plane and a sphere...

    I - A point divides a line into two parts; II - A line divides a plane into two parts; III - Does a smaller sphere divide a larger sphere into two parts, like layers of an onion? Note that the first two statements, the question of infinity must be considered. For the third statement, is the...
  5. Trysse

    I Prove that the geometric mean is always the same

    Given are a fixed point ##P## and a fixed circle ##c## with the radius ##r##. Point ##P## can be anywhere inside or outside the circle. I now draw two arbitrary lines ##l_1## and ##l_2## through the point ##P## in such a way, that both lines intersect with the circle ##c## in two distinct...
  6. A

    Converging geometric series

    I do not have any reasonable attempts at this problem, as I am trying to figure out how one can get the correct answer when we are not given any values. Maybe if some of you sees a mistake here, that implies that the values from the previous example should be used... ##a_3 = a_1 \cdot k{2}##...
  7. PainterGuy

    Medical How does human eye decode different geometric shapes such as circle?

    Hi, How does a human eye classify any shape as a circle, square, triangle etc.? Let's focus on a circular shape. Suppose we have a circle drawn in white on a black surface. The light falls on the retinal cells. I think the light falling on the retina will constitute a circular shape as well...
  8. H

    Vectors as geometric objects and vectors as any mathematical objects

    In geometry, a vector ##\vec{X}## in n-dimensions is something like this $$ \vec{X} = \left( x_1, x_2, \cdots, x_n\right)$$ And it follows its own laws of arithmetic. In Linear Analysis, a polynomial ##p(x) = \sum_{I=1}^{n}a_n x^n ##, is a vector, along with all other mathematical objects of...
  9. chwala

    Find the two values of the common ratio- Geometric sequence

    *kindly note that i do not have the solutions ... I was looking at this, not quite sure on what they mean by exact fractions, anyway my approach is as follows; ##\dfrac {a}{243}=\dfrac{a(1-r^3)}{240}## ##\dfrac {1}{243}=\dfrac{1-r^3}{240}## ##\dfrac {240}{243}=1-r^3##...
  10. samy4408

    I What does "upright" mean in geometric optics?

    if someone want to explain to me what is an upright image ? , and what are the other adjectives to define an image in geometric optics and their meaning , Thanks .
  11. G

    I Geometric Interpretation of Turbulence

    I would like to give a geometric interpretation to turbulence. Let's take into consideration for example a Poiseuille flow. The velocity profile resembles a parabolic bullet. As the particles are pushed by other layers of particles, then it must be that in addition to their translation, they...
  12. ergospherical

    Applied Lectures on the geometric anatomy of theoretical physics

    I stumbled across this series of 28 lectures by Dr Frederic Schuller of the university of Twente whilst searching for lectures about Lie theory. Having watched through lectures 13 to 18, I think they are simply superb (of course I'm assuming the rest are of similar quality). I only wish he would...
  13. D

    I Geometric Point of View of sets

    A set is nothing more than a collection. To determine whether or not an object belongs to the set , we test it against one or more conditions. If it satisfies these conditions then it belongs to the set, otherwise it doesn't. The geometric point of view of sets- a set can be viewed as being...
  14. C

    Proving geometric sum for complex numbers

    I went ahead and tried to prove by induction but I got stuck at the base case for ## N =1 ## ( in my course we don't define ## 0 ## as natural so that's why I started from ## N = 1 ## ) which gives ## \sum_{k=0}^1 z_k = 1 + z = 1+ a + ib ## . I need to show that this is equal to ## \frac{1-...
  15. A

    Straight line intersects geometric sequence

    Summary:: Two parallel lines (same slope) - one intersects the y-axis, and the other doesn't. Trying to find the intersection of either with a given geometric sequence. The lines are: y=mx y=mx+1 The values on one or the other of the lines - but not both simultaneously - are to be completely...
  16. J

    I Are Geometric Points Affected By Forces?

    Yesterday I found a playlist of videos by a youtuber "Dialect" who made a distinction between what he called Tier 1 and Tier 2 arguments of Relativity. Tier 2 promoted a view that acceleration was an observer dependent phenomena. In particular he was discussing the Twin Paradox, and he said...
  17. A

    A What is the point of geometric quantization?

    I studied the basics of geometric quantization for a recent work in quantum-classical hybrid systems1. It was an easy application of the method of gometric quantization (prequantization + polarization in ##\mathbb{R}^{3}##). The whole topic seems interesting since I want to learn more of...
  18. M

    Geometric Construction (bisecting an angle with a compass and straightedge)

    In discussing flight mechanics with a (15 years younger) co-worker with a doctorate in Aerospace Engineering. We examined some angles and I happened to mention bisecting an angle. I told him in High School in the early 1970's we learned how to bisect an angle with compass, and straightedge...
  19. pairofstrings

    B Arithmetic progression, Geometric progression and Harmonic progression

    How do I build functions by using Arithmetic Sequence, Geometric Sequence, Harmonic Sequence? Is it possible to create all the possible function by using these sequences? Thanks!
  20. N

    Computational Geometric Proofs Textbook

    I am seeking a geometry proofs textbook. In other words, I seek a textbook that shows all geometric proofs from start to finish. There are books that show proofs worked out as a reference book for students. Can someone provide me with a good geometry book for this purpose? I am particularly...
  21. T

    Mysteries of Geometric Optics In MTW Chapter 22

    At the start of this section §22.5 (Geometric Optics in curved Spacetime), the amplitude of the vector potential is given as: A = ##\mathfrak R\{Amplitude \ X \ e^{i\theta}\} ## The Amplitude is then re=expressed a "two-length-scale" expansion (fine!) but it then is modified further to...
  22. K

    What is the geometric approach to mathematical research?

    I read this article History of James Clerk Maxwell and it talks about Maxwell and Dirac also at some point. It is said that Maxwell thought geometrically, and also Dirac said he thought of de Sitter Space geometrically. They say their approach to mathematics is geometric. I see this mentioned...
  23. M

    A Finding Geometric Answers: Solving for n>7

    In a book (1984) with an interview of Coxeter, an old geometry question was described. Place a circle on a (2-d) lattice so that n points of the lattice are on the circumference. The answer for n=7 was given. Center is ##(\frac{1}{3},0)## and radius is ##\frac{5^8}{3}##. Has it been solved...
  24. Armine

    Proof of a formula with two geometric random variables

    The image above is the problem and the image below is the solution I have tried but failed.
  25. JD_PM

    I If L is diagonalizable, algebraic & geometric multiplicities are equal

    Given a n-dimensional vector space ##V## (where n is a finite number) and a linear operator ##L## (which, by definition, implies ##L:V \to V##; reference: Linear Algebra Done Right by Axler, page 86) whose characteristic polynomial (we assume) can be factorized out as first-degree...
  26. A

    A Convolution of two geometric distributions

    I'm trying to derive the convolution from two geometric distributions, each of the form: $$\displaystyle \left( 1-p \right) ^{k-1}p$$ as follows $$\displaystyle \sum _{k=1}^{z} \left( 1-p \right) ^{k-1}{p}^{2} \left( 1-p \right) ^{z-k-1}.$$ with as a result: $$\displaystyle \left( 1-p \right)...
  27. karush

    MHB 311.1.5.17 geometric description

    $\tiny{311.1.5.17}$ Give a geometric description of the solution set. $\begin{array}{rrrrr} -2x_1&+2x_2&+4x_3&=0\\ -4x_1&-4x_2&-8x_3&=0\\ &-3x_2&-3x_3&=0 \end{array}$ this can be written as $\left[\begin{array}{rrr|rr}-2&2&4&0\\-4&-4&-8&0\\&-3&-3&0\end{array}\right]$...
  28. anemone

    MHB Prove Geometric Sequence with $(a,b,c)$

    Let $a,\,b,\,c$ be non-zero real numbers such that $(ab+bc+ca)^3=abc(a+b+c)^3$. Prove that $a,\,b,\,c$ are terms of a geometric sequence.
  29. M

    Geometric figure drawing

    Are there free programs to add drawings to a post?
  30. fresh_42

    Changing the Statement Proving $\zeta(2)=\frac{\pi^2}{6}$ via Geometric Series & Substitutions

    Prove $$ \zeta(2) = \sum_{n\in \mathbb{N}}\dfrac{1}{n^2} = \dfrac{\pi^2}{6} $$ by evaluating $$ \int_0^1\int_0^1\dfrac{1}{1-xy}\,dx\,dy $$ twice: via the geometric series and via the substitutions ##u=\dfrac{y+x}{2}\, , \,v=\dfrac{y-x}{2}##.
  31. AN630078

    Geometric Sequence to solve an Interest Problem

    To find how much would be in the account after ten years, let the balance in the account at the start of year n be bn. Then b1=2000 I believe that this a compound interest problem. Common ratio r = 1.06 bn =2000*1.06^n−1 Thus, b10 =2000×1.06^9 = £3378.95791 The balance of the account at the...
  32. AN630078

    Geometric Sequence and the Limiting Value

    1. When n=1, u1+1=3-1/3(u1) u2=3-1/3(3) u2=2 When n=2 u2+1=3-1/3(u2) u3=3-1/3(2) u3=7/3 When n=3 u3+1=3-1/3(u3) u4=3-1/3(7/3) u4=20/9 The common ratio is defiend by r=un+1/un, but this is different between the terms, i.e. u2/u1=2/3 whereas u3/u2=(7/3)/2=7/6 Have I made a mistake? 2. A...
  33. K

    MHB Summation and geometric sums

    Hey! I'm stuck again and not sure how to solve this question been at it for a few hours. Any help is appreciated as always. Q: (1) Let the sum S = 3- 3/2 + 3/4 - 3/8 + 3/16 - 3/32 +...- 3/128. Determine integers a , n and a rational number k so that...(Image) (2 )And then calculate S using...
  34. U

    I Conditional distribution of geometric series

    Can someone help me on this question? I'm finding a very strange probability distribution. Question: Suppose that x_1 and x_2 are independent with x_1 ~ geometric(p) and x_2 ~ geometric (1-p). That's x_1 has geometric distribution with parameter p and x_2 has geometric distribution with...
  35. jisbon

    Geometric Series of nr^n

    So this seems to be a geometric Series, but with the coefficients in front, how do I exactly go about proving this? Thanks
  36. T

    I Dot product in Euclidean Space

    Hello As you know, the geometric definition of the dot product of two vectors is the product of their norms, and the cosine of the angle between them. (The algebraic one makes it the sum of the product of the components in Cartesian coordinates.) I have often read that this holds for Euclidean...
  37. J

    Energy of translation compared to the energy of rotation

    I use an example with a rack and a pinion. I suppose there is no losses from friction. I suppose the masses very low to simplify the study, and there is no acceleration. I suppose the tooth of the pinion and the rack perfect, I mean there is no gap. There is always the contact between the rack...
  38. K

    A Differential Forms or Tensors for Theoretical Physics Today

    There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor...
  39. Physics lover

    Longest geometric progression that can be obtained from a given set

    I am searching for an easy solution to such questions.I have been playing with it for few hours.I can only make a guess because I don't know how to solve such type of questions.Although I tried assuming first term as 'a',common difference as 'r'.And then the last term that is 'arn-1'should be...
  40. M

    A Eric WEinstein's Geometric Unity theory

    Eric Weinstein finally released a video of his 2013 Oxford talk on "geometric unity". There are many fans and skeptics out there, looking in vain for a genuinely informed assessment of the idea. I admit that so far I have only skimmed the transcript of the video, being very pressed for...
  41. S

    Engineering Kinematic and geometric similarity (fluids)

    My attempt at a solution is to start off first denoting V_a to be the automobile an V_e to be the economy version. Same goes with l_a and l_e. To try and relate the two I have tried: V_a I_a = V_l L_e, however I am really not sure how they got the square root. The answer is: v = V sqrt(l/L)...
  42. M

    MHB Which is the geometric interpretation?

    Hey! :o Which is the geometric interpretation of the following maps? $$v\mapsto \begin{pmatrix} 0&-1&0\\ 1&0&0\\ 0&0&-1\end{pmatrix}v$$ and $$v\mapsto \begin{pmatrix} 1& 0&0\\0&\frac{1}{2} &-\frac{\sqrt{3}}{2}\\ 0&\frac{\sqrt{3}}{2}&\frac{1}{2}\end{pmatrix}v$$
  43. M

    MHB Geometric interpretation of maps

    Hey! :o We have the below maps: $f_1:\mathbb{R}^2\rightarrow \mathbb{R}^2, \ \ \begin{pmatrix}x \\ y\end{pmatrix}\mapsto \begin{pmatrix}-x \\ -y\end{pmatrix}$ $f_2:\mathbb{R}^3\rightarrow \mathbb{R}^3, \ \ \begin{pmatrix}x \\ y\\ z\end{pmatrix}\mapsto \begin{pmatrix}x \\ -y\\...
  44. U

    Geometric Distribution

    I know the p.g.f. of X is $$q/(1-ps)$$ and that the mean is $$p/q$$, but how do I find a specific value for p here?
  45. G

    MHB Arithmetic string equal to the geometric string, count x

    Let x1, ..., x25 be such positive integers that x1⋅x2⋅ ... ⋅x25 = x1 + x2 + ... + x25. What is the maximum possible value of the largest of numbers x1, x2, ..., x25?
  46. F

    Geometric sum using complex numbers

    Solution to the problem tells us that ##S_5 + i S_6## is the sum of the terms of a geometric sequence and thus the solutions should be : $$S_5 = \frac{\sin( (n+1) x)}{\cos^n(x) \sin(x)},\,\,\,\, S_6 = \frac{\cos^{n+1}(x) - \cos((n+1)x)}{\cos^n(x) \sin(x)} , x \notin \frac{\pi}{2} \mathbb{Z}$$...
  47. U

    Finding the sum of a geometric series

    I'm using the sum of a geometric series formula, but I'm not sure how to find the ratio, r. The n is confusing me. The solution is below, but I'm having trouble with the penultimate step.
  48. S

    I Show isometry and find geometric meaning

    The matrix ##A## in question is ##\dfrac{1}{3} \left(\begin{array}{rrr} -2 & 1 & -2 \\ -2 & -2 & 1 \\ 1 & -2 & -2 \end{array}\right)## One can easily verify that ##AA^t=I##, hence an isometry. To find its geometric meaning, one can proceed to find ##U=\text{ker} \ (F-I)=\text{ker} \...
  49. S

    Understanding Sum to Infinity in Geometric Progression

    My question is Why is the sum to infinity used as opposed to Sum to n? and How can I deduce that the sum to infinity must be used from the question? Total Distance = h + 2*Sum of Geometric progression (to infinity) h + 2*h/3 / 1-1/3 h + 2h/3 *3/2 = h + h = 2h At first I did sum to infinity...
  50. christang_1023

    I How to understand this property of Geometric Distribution

    There is a property to geometric distribution, $$\text{Geometric distribution } Pr(x=n+k|x>n)=P(k)$$. I understand it in such a way: ##X## is independent, that's to say after there are ##(n+k-1)## successive failures, ##k## additional trials performed afterward won't be impacted, so these ##k##...