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 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 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. 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. 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. 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. 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. 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. 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. 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...