tiling – Can all quadrilaterals and triangles tessellate the Hyperbolic plane?

It is well known that in Euclidean geometry, all triangles and all quadrilaterals tessellate the plane. In non-Euclidean geometry, are there triangles and quadrilaterals that cannot tile the hyperbolic plane? If so, which triangle(quadrilateral) gives worst packing fraction?

Which convex region gives worst packing in hyperbolic plane?

Note: All these questions can be asked in elliptic geometry as well. One can also ask about the worst polygons for covering in the two geometries.

dnd 5e – True Seeing into Ethereal plane in Fog Cloud, what is seen?

True Seeing spell allows, among other things, this:

and can see into the Ethereal Plane, all out to a range of 120 feet.

Assume that True Seeing doesn’t allow one to directly see through the Fog Cloud fog, which is actual conjured fog and not an illusion, and creates a heavily obscured area. I believe this is a common ruling, not a contested topic, and anyway this whole question becomes moot otherwise.

But, Fog Cloud doesn’t say it extends to the Ethereal Plane. So, considering above, what does the creature under True Seeing effect see in a fog cloud? Most importantly, does this give any advantage, be it for fighting or moving or anything, over other creatures in the Fog Cloud?

Or, is it simply so, that what ever may be seen on the Ethereal Plane does not reflect what is going on in the Material Plane? Or the fog in fact does heavily obscure the Ethereal Plane, too? Or what?


I suppose related matter is, how are permanent obstacles like walls seen with True Seeing on the Ethereal Plane. My understanding is that they exist as “ethereal material” there, but I am not sure. Fog Cloud creates a temporary spell effect on the Material Plane though, so possibly different.

If this question can’t be answered within 5e material, drawing from earlier edition rules or lore is ok.

air travel – Can I take sealed PC parts in the plane from UK to Morocco?

I recently bought PC parts that cost me £4500 and I’m a student in UK for almost 3 years and I’m going back to my country which is Morocco in October.

Now i’m wondering if I am allowed to take these sealed PC parts with me, and if am allowed so there any tax should I pay, and if there is how much will be.

calculus and analysis – Contour integral around complex plane

I have this integral where it is singular at the point $p=0$, where I want to evaluate it on the solid contour given in the image below. My final goal is to find the integral in the range $(0,infty)$. My strategy is to use the solid contour and close the contour using the dashed lines then since there are no poles inside the closed contour, the integral in the closed contour is zero. Since the vertical dashed line contribution to the integral vanishes as $p=R rightarrow infty$, the integral in the range $(0,infty)$ is just negative times the rest of the contour (except the bottom solid contour which is what I want).

My problem is I don’t know how to find the integral of the upper dashed line plus the vertical solid line using my code. The small circular arc can be computed by hand using residues so that is no problem.

I know the integral along the whole solid line (except the circular arc) can be done using principal values but that does not allow me to find only the integral for the bottom solid line. Besides, I also don’t know how to find the principal value of the integral if one part of the domain is imaginary and the other part is real.

d = 2;
func(p_) := 1/(Cosh(p/2)^(2/d) Tanh(p/2) Sqrt(1 - (Cosh(p/2)^(4 - 4/d) Tanh(p/2)^2)/(-0.419602)))

Image

rotation – Vectors do not define a plane when constructing a RotationMatrix

Edit

Another way is use perturbation.

u = {0, 1, 0};
v = {0, -1, 0};
m = Limit(RotationMatrix({u, v + t*RandomInteger(20, 3)}), t -> 0)

{{0, 0, -1}, {0, -1, 0}, {-1, 0, 0}}

{{-(87/425), 0, -(416/425)}, {0, -1, 0}, {-(416/425), 0, 87/425}}

etc.

Original

We can find an orthogonal matrix satisfied $m^{T}bullet m=mathrm{Id},mathrm{Det}(m)=1$ which transform $(0,1,0)$ to $(0,-1,0)$

m = Array(a, {3, 3});
sol = Solve({Transpose(m) . m == IdentityMatrix(3), 
     Det(m) == 1, {0, 1, 0} . m == {0, -1, 0}})((1));
m /. sol
({0, 1, 0} . m /. sol) == {0, -1, 0}

{{a(1, 1), 0, -Sqrt(1 - a(1, 1)^2)}, {0, -1, 0}, {-Sqrt(1 - a(1, 1)^2), 0, -a(1, 1)}}

True

For example we can set a(1,1)->1/2

{{1/2, 0, -(Sqrt(3)/2)}, {0, -1, 0}, {-(Sqrt(3)/2), 0, -(1/2)}}

euler lagrange equation – acceleration of frictionless plane with mass sliding down the plane (Introduction to classical mechanics by Morin)

I am looking at this problem in Morin’s Introduction to classical mechanics:

6.1. Moving plane A block of mass $m$ is held motionless on a frictionless plane of mass $M$ and angle of
inclination $θ$ (see Fig. 6.8). The plane rests on a frictionless horizontal surface. The
block is released. What is the horizontal acceleration of the plane?

Here is the more detailed diagram associated with the solution:

enter image description here

The proposed solution sets out the Lagrangian in terms of the kinetic energy

$$
K=frac12Mdot q_1^2 + frac12 m ((dot q_1 + dot q_2)^2tan^2(theta) + dot q_2^2)
$$

and gravitational potential

$$
V=-mg(q_1+q_2)tan(theta)
$$

where I have replaced $x_i$ in the diagram with $q_i$.

This yields the Lagrangian

$$
L=K-V=frac12Mdot q_1^2 + frac12 m ((dot q_1 + dot q_2)^2tan^2(theta) + dot q_2^2)+mg(q_1+q_2)tan(theta)
$$

which, when processed with the Euler-Lagrange equations gives a solution (‘after a little simplification’)

$$
ddot q_1=frac{mgsin(theta)cos(theta)}{M+msin^2(theta)}
$$

(I did my best to process it with Sympy in python and got that solution as

$$
ddot q_1=frac{mgtan(theta)}{M tan^2(theta) + M + mtan^2(theta)}
$$

If you can see why the Sympy answer is equivalent let me know, but I will not be surprised if I typo’d something. I’m content to take the book’s solution as a given.)

To crosscheck things I also wanted to apply the Hamiltonian procedure, and so after replacing the $dot q_1=frac{p_1}{M}$ and $dot q_2=frac{p_2}{m}$

I had the Hamiltonian
$$
H=K+V=-mg(q_1 + q_2)tan(theta) + frac12 m ((p_2/m + p_1/M)^2tan^2(theta) + p_2^2/m^2) + frac12 p_1^2/M
$$

But processing this with Hamilton’s equation $dot p=-frac{partial H}{partial q}$

I think that should mean that $frac{ddot q_1}{M}=dot p_1=mgtan(theta)$ which (I think?) is not the same as the one suggested in the solution in the book. (It looks like the acceleration one would expect if the wedge were held fixed, anyhow, but I think that’s not what we should expect here.)

So at the moment (no pun intended) I’m stumped as to what I’m doing wrong. Can I not use the time derivative of the momentum to check the acceleration like this?

dg.differential geometry – Immersion of a part of the hyperbolic plane in $mathbb{R}^3$

I know that the pseudosphere is a regular surface with Gaussian curvature $-1$ that is not complete, also this surface is not complete. Hilbert’s theorem ensures that there is no isometric immersion of the hyperbolic plane $mathbb{H}^2$ in $mathbb{R}^3$, but if we remove a point from the hyperbolic plane, can it be immersed isometrically in $mathbb{R}^3$? Who could it be?

plotting – how to obtain a plane projection of a `StreamPlot3D`

StreamPlot3D are rather difficult to decipher. Is there a simple way to obtain the projection of such a plot onto a given plane? Ideally, I am looking for a method which does not even create the StreamPlot3D, since I am still in the process of getting the version 12.3 where this command appeared.

Draw A solid shape bounded by A rotating paraboloid z=x^2+y^2 and plane z=4, and find its projection region on plane Oxy

I figured out that the projected area is S<= 4pi, but the answer is x^2+y^2 <= 4, I think I misunderstood the projected area, I misunderstood the projected area as an area, okay, so if the projected area is not an area, it’s a space, what’s the space? Why is it x^2+y^2 <= 4?

linear algebra – Distance from a point to a plane (with matrices)

Let $W$ be some inner product space over $mathbb{R}^2$ where the inner product $langle,A,Brangle=tr(AB^t)$ ($B^t$ in this case is $B$ transposed).

$w_1=big(begin{smallmatrix}1&1\1&1end{smallmatrix}big)$, $w_2=big(begin{smallmatrix}0&1\1&0end{smallmatrix}big)$
and also $W=span{w1,w2}$,

Calculate the distance of
$v=big(begin{smallmatrix}4&0\4&4end{smallmatrix}big)$
from $W$.

I know that I need to first find the projection of $v$ onto $W$, and from there it’s smooth sailing. But, I don’t know how to represent W. In similar questions we usually want to find the Normal of the plane, and project $v$ onto it. However in this case, I don’t know how to find the normal of a span of matrices. We’d have to find a vector $U$ that is orthogonal to both $w_1,w_2$. I tried to find the orthonormal basis $B=(e_1,e_2)$, and say that $U=e_1+e_2$, but my $U$ came out to be equal to $w_2$ which didn’t make much sense.

Is there something I’m missing here?