How to compute the max/min surface area of a donut-shape solid generated by a revolved 2D circle, as the volume of the solid doesn’t change?

A donut shape solid is generated by revolving a circle $ (x-a)^2+y^2=b^2$ around the y-axis. $ a$ is the distance from the center of the hole of the donut to the center of the circle revolved, and $ b$ is the radius of the circle revolved. I’m trying to compute the minimized surface area and the maximized surface area (if the solid has) with the value of $ a$ and $ b$ , while the volume of the solid doesn’t change (which is $ 90\pi^2$ ). Thanks a lot if someone can help me 🙂

How to arbitrarily specify a face of planar graph as an external surface and draw it?

I learned this theorem in the graph theory textbook.

Theorem Every $ 2$ -connected plane graph can be embedded in the plane so that any specified face is the exterior.

G=PlanarGraph[{1 <-> 2, 1 <-> 3, 1 <-> 4, 2 <-> 3,               3 <-> 4, 2 <-> 5, 5 <-> 6, 6 <-> 3},               VertexLabels -> All] 

enter image description here

In the above embedding of this graph, we know $ 1256341$ is boundary exterior face of $ G$ .

I don’t know if there is a way to make the triangle face $ \Delta_{134}$ outside.

The above is just an example. For the graph $ G$ , maybe I can change the layout of some points by VertexCoordinates. But for the large number of vertices, I don’t know if there is a good and unified way to arbitrarily specify an external face and give a good plane drawing.

3D Arrow path along a surface

I have the following two surfaces, created by the following functions

En1[\[Delta]_,g1_,g2_,k_]:=1/2(-I g1+I g2-Sqrt[-(g1+g2-2k+I \[Delta])(g1+g2+2k+I\[Delta])]) En2[\[Delta]_,g1_,g2_,k_]:=1/2(-I g1+I g2+Sqrt[-(g1+g2-2k+I \[Delta])(g1+g2+2k+I\[Delta])])  g1 = 2;  g2 = -3;  \[Delta]max = 2;  kmax = 1;   a = Plot3D[{Re[En1[\[Delta],g1,g2,k]],Re[En2[\[Delta], g1, g2,k]]},{\[Delta],-\[Delta]max,\[Delta]max}, {k, 0, kmax},AxesLabel->{"\[Delta]", "g"}, PlotStyle->Opacity[0.7],BoxRatios -> {1, 1, 1}];  start = Graphics3D[{Black,Ellipsoid[{-1,0.8,Re[En1[-1,g1,g2,0.8]]},{0.1,0.025,0.075}]}];  end = Graphics3D[{Black,Ellipsoid[{-1,0.8,Re[En2[-1,g1,g2,0.8]]},{0.1,0.025,0.075}]}];   test1=Show[a, start, end, PlotRange -> All] 

enter image description here

Now I would like to make a 3D arrow path by going, along the surface, from one black point to the other one, encircling the origin $ (\delta,k)=(0,0.5)$ .

It seems Mathematica do not have a way to create a path, so I constructed it through a ParametricPlot.

test2 = ParametricPlot3D[{-Cos[t],0.5+kmax Sin[t],Re[En1[-\[Delta]max Cos[t], g1, g2, 0.5 + kmax Sin[t]]]},{t,0,2\[Pi]}];   Show[test1, test2] 

enter image description here

However it seems that this it is not encircling nor being created on the surface. Any thoughts?

Can Tentacle of the Deeps be cast on the surface of water?

The Warlock subclass The Fathomless has an ability called Tentacle of the Deeps that reads:

You can magically summon a spectral tentacle that strikes at your foes. As a bonus action, you create a 10-foot-long tentacle at a point you can see within 60 feet of you. The tentacle lasts for 1 minute or until you use this feature to create another tentacle. When you create the tentacle, you can make a melee spell attack against one creature within 10 feet of it. On a hit, the target takes 1d8 cold damage, and its speed is reduced by 10 feet until the start of your next turn. When you reach 10th level in this class, the damage increases to 2d8. As a bonus action on your turn, you can move the tentacle up to 30 feet and repeat the attack. You can summon the tentacle a number of times equal to your proficiency bonus, and you regain all expended uses when you finish a long rest.

Would that be able to stand on the surface of water?

Does Detect Magic make an Arcane Mark placed on an invisible surface glow?

Invisible arcane marks glow when hit by Detect Magic.

Usually, how this works is that someone casts Detect Magic on an item and they see this fluorescent drawing on it, marking it as the belonging of some specific mage.

This time, however, we have a creature that likes turning invisible every time it hears our magus cast True Strike or every time too many people swarm them, and the magus would like to slap an arcane mark on their back so that, when the creature goes invisible, a round 1 Detect Magic causes the glow, marking the square and letting everyone but the Detect Magic caster target the right square.

Does this work?

Does “a point you choose” include any movable surface?

I read somewhere that spells like Darkness and Silence, where the effect of the spell spreads from a point I choose within range, can be cast on the surface of a moveable item. But after extensive googling, I couldn’t find confirmation on it anymore. These are the relevant parts of the spell descriptions:

Darkness
Magical darkness spreads from a point you choose within range to fill a 15-foot radius sphere for the duration. …

Silence
For the duration, no sound can be created within or pass through a 20-foot-radius sphere centered on a point you choose within range. …

So theoretically, does this mean that a Warlock with the Devil’s Sight incantation can cast Darkness on their armor and "carry" it around with him? Giving him semi-permanent (Concentration) advantage to targets without Devil’s Sight/Truesight?

Another example would be someone casting Silence on an enemy mage, rendering them unable to cast spells with Verbal components also semi-permanently (Concentration)?

Converting a 2d surface into a 3d volume

I have a 2d surface shown in the image below using ListPlot3D[]:

image It is generated with one line of code:

ListPlot3D[RandomVariate[UniformDistribution[], {10,10}]]     

Currently, it’s a ‘white noise’ surface, meaning the surface is more or less random between zero and one. I’d like to give this thickness (so I can ultimately 3d print it).

I’d like the remaining 5 sides to be flat, so that I’d end up with a "cube" with one surface jagged.

Is there an easy way to do this? Essentially just fill in everything below the surface?

Thanks!