Questions tagged [geometry]

To be used for questions on geometry closely pertaining to physics. Includes differential geometry and euclidean geometry.

Filter by
Sorted by
Tagged with
151
votes
2answers
29k views

Why do sunbeams diverge even though the sun is much more than a few kilometers away?

Consider this picture of sun beams streaming onto the valley through the clouds. Given that the valley is only (at a guess) 3km wide, with simple trigonometry and the angles of the beams, this gives ...
146
votes
9answers
23k views

Why are the wet patches on these floor tiles circular?

My friend's 3-year-old daughter asked "Why are there circles there?" It had either rained the night before or frost had thawed. What explains the circles? Follow-up question: Ideally, are these ...
88
votes
4answers
17k views

Seeing something from only one angle means you have only seen (what?)% of its surface area at most?

Is there a logical/mathematical way to derive what the very maximum percentage of surface area you can see from one angle of any physical object? For instance, if I look at the broad side of a piece ...
65
votes
8answers
113k views

What is the physical significance of dot & cross product of vectors? Why is division not defined for vectors?

I get the physical significance of vector addition & subtraction. But I don't understand what do dot & cross products mean? More specifically, Why is it that dot product of vectors $\vec{A}...
47
votes
8answers
8k views

Why do we use cross products in physics?

We can define cross products mathematically like if we take two vectors, we can find another vector with certain properties but why do we use it in physics, if we consider a hypothetical physical ...
40
votes
7answers
8k views

Why does the full Moon appear?

I know that the full Moon appears when Sun, Moon and Earth are in a straight line, but if we consider that they are in straight line, why is the Moon illuminated? I mean to say that Earth should ...
33
votes
11answers
8k views

Is the “spacetime” the same thing as the mathematical 4th dimension?

Is the "spacetime" the same thing as the mathematical 4th dimension? We often say that time is the fourth dimension, but I am wondering if it's means that time is like the fourth geometrical axis, or ...
26
votes
9answers
10k views

How to interpret the units of the dot or cross product of two vectors?

Suppose I have two vectors $a=\left(1,2,3\right)$ and $b=\left(4,5,6\right)$, both in meters. If I take their dot product with the algebraic definition, I get this: $$a \cdot b = 1\mathrm m \cdot 4\...
22
votes
4answers
31k views

Distinguishing between solid spheres and hollow spheres (equal mass)

If there are two spheres (hollow and solid) with equal mass and radius and we want to find the hollow sphere without using any equipment. What's the best way(s) to recognize the hollow sphere and ...
22
votes
4answers
176k views

Does the rotation of the earth dramatically affect airplane flight time?

Say I'm flying from Sydney, to Los Angeles (S2LA), back to Sydney (LA2S). During S2LA, travelling with the rotation of the earth, would the flight time be longer than LA2S on account of Los Angeles ...
22
votes
7answers
8k views

Experimental evidence of a fourth spatial dimension?

As human beings, we observe the world in which we live in three dimensions. However, it is certainly theoretically possible that more dimensions exist. Is there any direct or indirect evidence ...
21
votes
1answer
3k views

Land proportions in NASA blue marble photographs

What is the explanation for the apparent size difference of North America in these two photos from NASA? Image source Image source
21
votes
3answers
9k views

Why can't a piece of paper (of non-zero thickness) be folded more than $N$ times?

Updated: In order to fold anything in half, it must be $\pi$ times longer than its thickness, and that depending on how something is folded, the amount its length decreases with each fold differs. –...
19
votes
3answers
5k views

How can a full moon be seen south of an observer's location?

I know this seems like a simple question, but I'm trying to debate with a flat earth theorist. I asked him to explain why can the ISS visibly be seen orbiting the Earth with the naked eye, and he put ...
19
votes
9answers
10k views

Why do objects appear smaller when viewed from a distance? [duplicate]

Yes, I know all about perspective (I'm an artist). I even have some basic knowledge of descriptive geometry. I know how it works. My question is more about why it works. I have a sneaking suspicion ...
18
votes
3answers
3k views

Why do water drops form spheres in space? [duplicate]

When water is poured out in space, why does it always take a spherical ball-like shape?
18
votes
7answers
12k views

How big are clouds? [closed]

How big are clouds? When I look up into the sky I have no frame of reference, so I don't know if they are 200 feet or 2 miles across. When I am in a plane looking out at a cloud, I try to use the wing ...
18
votes
2answers
2k views

How does one measure space-like geodesics? Or: What is the physical interpretation of space-like geodesics?

In general relativity, time-like geodesics are the trajectories of free-falling test particles, parametrized by proper time. Thus, they are easy to interpret in physical terms and are easy to measure (...
17
votes
5answers
8k views

Does it contradict special relativity that an electron beam in a television picture tube can move across the screen faster than the speed of light?

While looking at some exercises in my physics textbook, I came across the following problem which I thought was quite interesting: It is possible for the electron beam in a television picture tube ...
17
votes
2answers
1k views

Generalized Complex Geometry and Theoretical Physics

I have been wondering about some of the different uses of Generalized Complex Geometry (GCG) in Physics. Without going into mathematical detail (see Gualtieri's thesis for reference), a Generalized ...
16
votes
2answers
1k views

Yang-Mills vs Einstein-Hilbert Action

The classical Yang-Mills action is of the form $$S=\frac{1}{2g^2}\int_{\mathcal{M}}\text{tr}\left[F\wedge\star F\right]\\ =\frac{1}{4g^2}\int\mathrm{d}^dx\sqrt{g}\,\text{tr}\left[F^{\mu\nu}F_{\mu\nu}\...
16
votes
2answers
2k views

The Reeh-Schlieder theorem and quantum geometry

There have been some very nice discussions recently centered around the question of whether gravity and the geometry and topology of the classical world we see about us, could be phenomena which ...
15
votes
11answers
1k views

Is there a proof that the set of real numbers can exactly represent distances? [duplicate]

Mathematicians define real numbers in an abstract way - as an 'ordered field' with 'the least upper bound property'. In physics, we use real numbers to represent distances. For us to be able to do ...
13
votes
4answers
4k views

How is Chasles' Theorem, that any rigid displacement can be produced by translating along a line and then rotating about the same line, true?

Chasles' Theorem in its strong form says: The most general rigid body displacement can be produced by a translation along a line (called its screw axis) followed (or preceded) by a rotation about ...
13
votes
5answers
10k views

Why can the cross product of two vectors be calculated as the determinant of a matrix?

The cross product $\vec{a} \times \vec{b}$ can be written as the determinant of the matrix: $$\left| \begin{matrix} \vec{i} & \vec{j} & \vec{k} \\ a_i & a_j & a_k \\ b_i & b_j &...
13
votes
0answers
338 views

The dual of a surface element in 4-space

In reading the classic text, "The Classical Theory of Fields", Third Edition, by Landau and Lifschitz, I found an "obvious" statement not so obvious to me. It is on p.19, the statement of the ...
12
votes
3answers
1k views

How many times can light revolve around a black hole?

Take a light ray approaching a black hole from infinity which goes out again to infinity. What is the maximum finite rotation it can describe? (I know it can loop around indefinitely in the ...
12
votes
3answers
7k views

How did Eratosthenes know the suns rays are parallel?

Eratosthenes famously observed that the suns rays were perpendicular to the ground in one location, yet non-perpendicular to the ground at a location some miles to the north. On the assumption that ...
12
votes
2answers
2k views

Is Dyson Sphere a stable construction?

Suppose that a star is encompassed by a Dyson Sphere. Do we need a position control system for the Dyson Sphere to keep its origin always aligned with the center of the star? Will it stay aligned ...
11
votes
5answers
1k views

Geodesics: Straightest or Shortest? When and Why?

In classical General Relativity (meaning not modified) one can think of geodesics in two ways. One way is to say that a geodesic is the curve which is the straightest (in analogy with the flat case) ...
11
votes
4answers
7k views

Why does the light side of the moon appear not to line up correctly with the evening sun?

I live at roughly $(52.4^\circ,-2.1^\circ)$. On sunny evenings I've often looked at the Moon and the Sun and noticed that the light part of the Moon does not appear to line up with the Sun. For ...
10
votes
11answers
4k views

Is it possible for a physical object to have a irrational length?

Suppose I have a caliper that is infinitely precise. Also suppose that this caliper returns not a number, but rather whether the precise length is rational or irrational. If I were to use this ...
10
votes
4answers
15k views

How far into space does one have to travel to see the entire sphere of earth?

Virgin Galactic will take passengers aboard SpaceShipTwo as high as 65 miles above the surface of the earth. But from this altitude, passengers will only be able to see a certain segment of the ...
10
votes
3answers
2k views

What is the exact meaning of homogeneity in cosmology?

I understand that, in general, homogeneity is the physical attribute of being uniform in composition (" of the same form at every point"), but I'm slightly confused when it is used in cosmology as ...
10
votes
3answers
1k views

In a ball with random thread/strings, how does the density of threads/strings change with radius?

A large plastic ball full of holes is given. (So the holes are in a plastic shell.) Straight threads connect these holes randomly, by passing through the interior of the ball/shell. For a big ball or ...
10
votes
2answers
861 views

Why are conformal transformations so prevalent in physics?

What is it about conformal transformations that make them so widely applicable in physics? These preserve angles, in other words directions (locally), and I can understand that might be useful. Also, ...
9
votes
9answers
9k views

Gravitation is not force?

Einstein said that gravity can be looked at as curvature in space- time and not as a force that is acting between bodies. (Actually what Einstein said was that gravity was curvature in space-time and ...
9
votes
3answers
446 views

Can we see the curvature of a surface?

After reading the Feynman lectures' (chapter 42, Vol.2) , it had me thinking if it is by any way possible to measure the curvature of a surface (think, surface of earth) just by observing the nature ...
9
votes
4answers
4k views

What's the dimensionality of a solid angle?

I haven't seen this explained clearly anywhere. Solid angles are described usually as a fraction of the surface area of a unit sphere, similar to how angles are the fraction of the circumference of a ...
9
votes
1answer
262 views

Cutting a circle and moving endpoints

A metal (or otherwise, suitably elastic) circle is cut and the points are slid up and down a vertical axis as shown: How would one describe the resultant curves mathematically?
9
votes
1answer
822 views

Understanding Calculus Notation in Physics

I have just started a first-year calculus-based physics course about electromagnetism and waves. I am having trouble understanding what calculus notation means in the context of physics. Here is a ...
9
votes
2answers
1k views

Geometrical interpretation of the Dirac equation

Is there an intuitive geometrical picture behind the Dirac equation, and the gamma matrices that it uses? I know the geometric algebra is a Clifford algebra. Can the properties of geometric algebra, ...
9
votes
1answer
575 views

Can masses move in 2+1 gravity?

I would like to understand basic concepts of the general relativity in 2+1 spacetime. As far as I know, GR predicts that such a spacetime is flat everywhere except for the point masses which create ...
8
votes
3answers
509 views

Question about associative 3-cycles on G2 manifolds

Let $X$ be a manifold with $G_2$ holonomy and $\Phi$ be the fundamental associative 3-form on $X$. Let $*\Phi$ be the dual co-associative 4-form on $X$. Now consider a particular associative 3-cycle $...
8
votes
2answers
4k views

Triple-right triangle experiment: what's the minimum distance?

As I had showed in a previous answer, among the other ways to prove that the Earth is round, we have the triple-right triangle. The idea is simple: Starting from point A you move in a straight line ...
8
votes
2answers
633 views

Why does the $L_2$ norm give the shortest path between 2 points?

Why not the $L_1$ or $L_3$ distances? Is there some deep reason why the universe (at least at human scales) looks pretty much Euclidean? Could we imagine a different universe where a different $...
8
votes
4answers
623 views

Physical representation of volume to surface area

I was looking at this XKCD what-if question (the gas mileage part), and started to wonder about the concept of unit cancellation. If we have a shape and try to figure out the ratio between the volume ...
8
votes
1answer
607 views

Why are fractal geometries useful for compact antenna design?

While most of what I've read about fractals has been dubious in nature, over the years, I keep hearing that these sorts of self-similar (or approximately self-similar) geometries are useful in the ...
8
votes
1answer
647 views

The role of metric in the Wave Equation

The wave equation is often written in the form $$(\partial^2_t-\Delta)u=0,$$ involving the Laplace-Beltrami operator $\Delta$. However, the Laplace-Beltrami operator $\Delta$ is defined only in the ...
7
votes
3answers
5k views

Can I calculate the size of a real object by just looking at the picture taken by a Camera?

Can I calculate the size of a real object by just looking at the picture taken by a Camera? (I think people do that) i dont understand how? (from physics point of view)

1 2 3 4 5 13