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.
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...
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
(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...
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...
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...
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}##...
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...
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...
*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##...
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 .
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...
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...
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...
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-...
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...
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...
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...
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...
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!
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...
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...
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...
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...
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...
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: $$\displaystyle \left( 1-p \right)...
$\tiny{311.1.5.17}$
Give a geometric description of the solution set.
$\begin{array}{rrrrr}
-2x_1&+2x_2&+4x_3&=0\\
-4x_1&-4x_2&-8x_3&=0\\
&-3x_2&-3x_3&=0
\end{array}$
this can be written as
$\left[\begin{array}{rrr|rr}-2&2&4&0\\-4&-4&-8&0\\&-3&-3&0\end{array}\right]$...
Prove
$$
\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}##.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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)...
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$$
Let x1, ..., x25 be such positive integers that x1⋅x2⋅ ... ⋅x25 = x1 + x2 + ... + x25. What is the maximum possible value of the largest of numbers x1, x2, ..., x25?
Solution to the problem tells us that ##S_5 + i S_6## is the sum of the terms of a geometric sequence and thus the solutions should be :
$$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}$$...
I'm using the sum of a geometric series formula, but I'm not sure how to find the ratio, r. The n is confusing me.
The solution is below, but I'm having trouble with the penultimate step.
The matrix ##A## in question is
##\dfrac{1}{3}
\left(\begin{array}{rrr}
-2 & 1 & -2 \\
-2 & -2 & 1 \\
1 & -2 & -2
\end{array}\right)##
One can easily verify that ##AA^t=I##, hence an isometry. To find its geometric meaning, one can proceed to find ##U=\text{ker} \ (F-I)=\text{ker} \...
My question is Why is the sum to infinity used as opposed to Sum to n? and How can I deduce that the sum to infinity must be used from the question?
Total Distance = h + 2*Sum of Geometric progression (to infinity)
h + 2*h/3 / 1-1/3
h + 2h/3 *3/2 = h + h = 2h
At first I did sum to infinity...
There is a property to geometric distribution, $$\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##...