dnd 5e – Is there any mundane way to travel from the Border Ethereal of a plane to the plane itself?

Say a commoner with no spells, class features, or magic items is Plane Shifted from the Material Plane to the Border Ethereal.

The DMG makes mention of “curtains” that allow travel from the Deep Etheral to the Border Ethereal, but nothing of any portals in the Ethereal plane itself.

Is there any way for this individual to return to the Material Plane?

What is the analytical form of the cylindrical wave appearing on reflection of a plane wave from a corner?

This is a cross-post from Math.SE, where no answer was given after 3 months.


Consider a plane 2D wavelet moving towards a corner reflector with 120° opening angle with infinitely extended sides. The surface of the reflector has homogeneous Dirichlet boundary conditions imposed, and the wave obeys the usual hyperbolic wave equation:

$$partial_{tt} u(x,y,t)=partial_{xx}u(x,y,t)+partial_{yy}u(x,y,t).$$

The solution, with the initially propagating part and the reflections, can be easily constructed by rotating the initial wavelet, changing its sign and putting the resulting wavelet next to the initial one so as to satisfy the boundary conditions by canceling the wave function at the boundaries. The result will look like this (ignore the small-wavelength artifacts, they are due to numeric errors in the simulation):

incident wave with the reflections

But as the points of the slanted reflected waves come close to the corner, there appears a problem: simply “sliding” the reflected part no longer works, since the shape of the reflecting boundary abruptly changes. Moreover, numerical simulation (see below) shows that the reflection from the corner doesn’t produce a backwards-propagating plane wave (that we’d get from a 90° corner): instead the original slanted reflections continue their paths, and a new, cylindrical, wave originates from the corner. This cylindrical wave appears to cancel out the values of the slanted reflections on the boundary to satisfy the boundary conditions.

reflected wave

My question is: what is the analytical form of this cylindrical wave? It doesn’t seem to be a Bessel function, because Bessel functions don’t have constant amplitude nor constant wavelength (they change with radius). So what is it then? Does it have a closed form? Or is it at least explicitly expressible as an integral or a series?

plotting – 3D plot of Intersection of sphere with plane (basic)

We use the implicit exprssion of plane. The normal of plane is Cross(b-a,c-a)

({x, y, z} - a).Cross(b - a, c - a)==0

And we also use the implicit expression of sphere,here {5,0,0} is the sphere center and 10 is radius.

Norm({x, y, z} - {5, 0, 0}) - 10==0

Norm({x, y, z} - {5, 0, 0}) - 10 as MeshFunction

x = InfiniteLine({{0, 0, 0}, {1, 0, 0}});
y = InfiniteLine({{0, 0, 0}, {0, 1, 0}});
z = InfiniteLine({{0, 0, 0}, {0, 0, 1}});
plane = InfinitePlane({{1/2, 0, 0}, {1/2, 1, 0}, {1/2, 0, 1}});


sphere = Sphere({5, 0, 0}, 10);
sphereOrigin = Point({5, 0, 0});

fig = Graphics3D({{Thick, x}, {Thick, y}, {Thick, z}, {Opacity(0.15), 
     plane}, {Opacity(0.15), sphere}, {PointSize(Large), Red, 
     sphereOrigin}}, Boxed -> False);

{a, b, c} = {{1/2, 0, 0}, {1/2, 1, 0}, {1/2, 0, 1}};
circle3 = 
  ContourPlot3D(({x, y, z} - a).Cross(b - a, c - a) == 0, {x, -15, 
    15}, {y, -15, 15}, {z, -15, 15}, 
   MeshFunctions -> 
    Function({x, y, z}, Norm({x, y, z} - {5, 0, 0}) - 10), 
   Mesh -> {{0}}, MeshStyle -> {Thick,Red}, ContourStyle -> None, 
   BoundaryStyle -> None);
Show(fig, circle3)

enter image description here

lens – Seeming contradiction on wikipedia about rear nodal points and how the principal plane is related to focal length

There are two lines on the wikipedia pages for “cardinal points” and “focal length” that seem to contradict each other, and I would be extremely grateful if someone could explain to me why they do not. In the page for cardinal points, it says:

If the medium surrounding the optical system has a refractive index of 1 (e.g., air or vacuum), then the distance from the principal planes to their corresponding focal points is just the focal length of the system. In the more general case, the distance to the foci is the focal length multiplied by the index of refraction of the medium.

This makes sense to me. I also get that these principal planes can often be located outside of the lens with some clever optics, allowing for lenses that are physically shorter than their focal length. However, on the page for focal length, the page reads:

When a photographic lens is set to “infinity”, its rear nodal point is separated from the sensor or film, at the focal plane, by the lens’s focal length. Objects far away from the camera then produce sharp images on the sensor or film, which is also at the image plane.

I don’t see how these can both be true, because if the focal point, the point as I understand it to be where all the light converges, was on the film plane, an image wouldn’t be rendered, it would just be an indistinguishable point of light. Does the light not have to travel a distance past the focal point to the film plane in order to form an image?

I think it is possible that I am getting my front and rear nodal points confused, or that I have a larger fundamental misunderstanding about how focal length is measured. Thank you so much for your help!

geometry – finding a plane that forms an acute angle $theta/2$ with two planes that have $theta$ as angle between them

Let $theta$ denote the angle between the planes $2x+2y+z = 1$ and $-x + 2y – 2z = 2$. The planes intersect in a line that we call L. Find the plane that contains L and forms acute angle $theta/2$ with both planes.

First of all, $n_1 = <2, 2, 1>$ and $n_2 = <-1, 2, -2>$. Then $cos theta$ gives that $theta = 90°$, using the dot product.Thus the angle the plane in question forms with the two planes would be $45°$ each.

I’m unsure about what to do next, however. Let $n_3$ be the normal to the plane the problem asks for—plane 3. Depending on how I draw the 3 planes, it seems that I could either get $n_3 = n_2 – n_1$ (or $n_3 = n_1 – n_2$; the key is $n_3$ is the difference of the normals of two given planes) or $n_3 = n_1 + n_2$. And computing in these two different ways give different planes. Which one would I know is the correct approach (without using any of the graphing calculators etc.)?

Or would both planes be accepted as the answer, since in this case $theta = 90°$? What if $theta$ was an acute or obtuse angle, however? How would my problem-solving approach change, to still get the plane that contains L and forms $theta/2$ (acute) angle with the two planes?

dnd 5e – Can a Way of Shadow monk use Shadow Step to teleport to the Ethereal Plane if they can see into it?

No, because Shadowstep is based on teleportation.

Ok, Robe of Eyes allows you to see into the Ethereal. So far, so good.

And Shadowstep says:

When you are in dim light or darkness, as a bonus action you can
teleport up to 60 feet to an unoccupied space you can see that is also in dim light or darkness.

Emphasis mine.

The Shadowstep movement is done through teleportation. If it was a separate way to transport, then the spell would say so. Instead is specifically says “teleport”. Thus, it is a form of teleportation. So we have to also check how the Teleport spell itself works, because any limitation on teleportation will also apply to Shadowstep too.

Teleport:

The destination you choose must be known to you, and it must be on the
same plane
of existence as you.

Pretty simple here: Despite seeing into the ethereal plane, the actual destination, while within Shadowstep’s apparent visible range, still remains on another plane, which teleportations cannot bring you to.

Note also that spells allowing planar travel explicitly do not use the word “teleport”.

dnd 5e – Can I shadowstep to the Ethereal Plane if I can see into it?

No, because Shadowstep is based on teleportation.

Ok, Robe of Eyes allows you to see into the Ethereal. So far, so good.

And Shadowstep says:

When you are in dim light or darkness, as a bonus action you can
teleport up to 60 feet to an unoccupied space you can see that is also in dim light or darkness.

Emphasis mine.

The Shadowstep movement is done through teleportation. If it was a separate way to transport, then the spell would say so. Instead is specifically says “teleport”. Thus, it is a form of teleportation. So we have to also check how the Teleport spell itself works, because any limitation on teleportation will also apply to Shadowstep too.

Teleport:

The destination you choose must be known to you, and it must be on the
same plane
of existence as you.

Pretty simple here: Despite seeing into the ethereal plane, the actual destination, while within Shadowstep’s apparent visible range, still remains on another plane, which teleportations cannot bring you to.

Note also that spells allowing planar travel explicitly do not use the word “teleport”.

lore – Is the negative energy plane underneath Golarion?

I had always been under the impression that Golarion and the negative energy plane were on entirely different metaphysical planes. This would mean that you couldn’t simply move from one to the other using mundane means.

The Tar-Baphon entry on Lost Omens: Legends says (pg.104):

Tar-Baphon dug a portal to the Negative Energy Plane on the Isle of Terror …

To me, this sounds like he physically dug a hole to the negative energy plane. At face value, this would only be possible if the negative energy plane were somehow underneath the earth of Golarion.

I have access to many of the PF2 books, but am weak in lore published in PF1 resources. Is it true that the negative energy plane is literally below Golarion?

Plotting line segments in complex plane

Recent versions of Mathematica have introduced various functions for plotting in terms of complex numbers and complex functions, including ComplexPlot, ComplexListPlot, ComplexRegionPlot, and ComplexVectorPlot. These functions allow direct use of complex numbers and complex functions without the user having to explicitly apply ReIm or otherwise split complex objects into their real and imaginary parts.

Question: Is there a built-in function that provides some very basic and essential plotting functionality for geometric objects specified in terms of complex numbers. For example:

  • plotting a line segment in the complex plane by directly specifying its endpoints as complex numbers?
  • plotting a circle (not a filled disk!) in the complex plane by specifying its center directly as a complex number and its radius?

(And if not, why on earth not?? Given that Mathematica regards complex numbers as being so fundamental that one must explicitly override assumptions of numbers being complex when you want them to be real, it seems surprising to me that it’s taken even this long for Mathematica to build in the complex plotting functions I cite.)

As a very basic and simple example, I want to do make the following kind of graphics — without having to use ReIm to split up explicitly all the complex numbers into their real and complex parts.

Graphics({Circle(ReIm(2 + 2 I), 1), PointSize(Large), Red, 
  Point(ReIm(2 + 2 I)), Thick, Blue, Line(ReIm({1 + 2 I, 3 + 2 I}))}, 
 Axes -> True, AxesOrigin -> ReIm(0))

(!(enter image description here)(1))(1)

At least ComplexListPlot would allow plotting one element there, namely, the center of the circle, and that could be combined using a Show with the other graphics. Still, that leaves the circle and line segment to treat.
(1): https://i.stack.imgur.com/MJBAq.png

dungeons and dragons – If a Defiler native to Athas managed to get to another Material Plane, could they still use Defiling magic?

From what I understand, in Dark Sun, Defilers can destroy the plants around to make their magic more powerful. If a Defiler from Athas somehow ended up in another material plane (don’t care which, don’t care how), could they use the abundant life energy to power their spells? Or does Defiling only work on Athas due to the Weave or something?