Tricky 3d geometry problem
Solution 1
If we place the cube with its main diagonal from $(0,0,0)$ to $(1,1,1)$ and three edges along the axes, then we can parametrize two edges and a diagonal: $$ \begin{align} edge_1&:s\mapsto\begin{pmatrix}s\\0\\0\end{pmatrix},\quad s\in[0,1]\\ edge_2&:s\mapsto\begin{pmatrix}1\\0\\s\end{pmatrix},\quad s\in[0,1]\\ diag&:x\mapsto\frac{1}{\sqrt 3}\begin{pmatrix}x\\x\\x\end{pmatrix},\quad x\in[0,\sqrt 3] \end{align} $$ For a given $s\in[0,1]$ one can minimize the quadratic expression (just pick the vertex) $$ diag(x)edge_1(s)^2 $$ with respect to $x$ to find that $s=\sqrt 3 x$ and with this the distance $f(x)$ between the point $diag(x)$ on the diagonal and the point $edge_1(s)$ on $edge_1$ is $$ f(x)=\sqrt 2 x $$ Similarly, one may deduce that for $$ diag(x)edge_2(s)^2 $$ to be minimized wrt. $x$ for a fixed $s\in[0,1]$ we must have $s=\sqrt 3 x1$ and so the distance $g(x)$ between the diagonal and $edge_2$ is $$ g(x)=\sqrt{2(x^2\sqrt 3x+1)} $$ By symmetry, we may conclude that the curve we are rotating is $$ h(x)= \begin{cases} \sqrt 2 x&\text{ for }x\leq\tfrac13\sqrt 3\\ \sqrt 2(x\sqrt 3)&\text{ for }x\geq \tfrac23\sqrt 3\\ \sqrt{2(x^2\sqrt 3x+1)}&\text{ in between} \end{cases} $$ defined on the domain $x\in[0,\sqrt 3]$ which is illustrated here:
Remark: Fixing $s$ and varying $x$ fixes a point on an edge and varies a point on the diagonal until the nearest point is found. Doing it the other way around would result in a wrong construction of fixing a point on the diagonal and finding the nearest point on the given edge, which minimizes distance orthogonal to an edge instead of orthogonal to the diagonal/axis of rotation.
To demonstrate how it fits, here is an overlay in a dynamic 3Dmodel of it:
The red curve is the function $h(x)$ derived above corresponding to the "union" case of the solid formed by the uncountable union of all positions of a full rotation of the cube. The purple lines describe the "intersection" case, the uncountable intersection of all positions in a full rotation of the cube.
Solution 2
Let's take the unit centered cube, with vertexes at $\pm 1$. To rotate it so that its main diagonal gets aligned with the $x$ axis (vertical axis in the figure) we can use two rotations along two axes, the first by 45 degrees, the second by $\tan^{1}(\sqrt{1/2})=\sin^{1}(\sqrt{1/3})$. We get then the rotation matrix:
$$ Q= \begin{pmatrix} \sqrt{\frac{1}{3}} & 0 & \sqrt{\frac{2}{3}} \\ 0 & 1 & 0 \\ \sqrt{\frac{2}{3}} & 0 & \sqrt{\frac{1}{3}} \\ \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ 0 & \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \end{pmatrix}= \begin{pmatrix} \sqrt{\frac{1}{3}} & \sqrt{\frac{1}{3}} & \sqrt{\frac{1}{3}} \\ 0 & \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ 2\sqrt{\frac{1}{6}} & \sqrt{\frac{1}{6}} & \sqrt{\frac{1}{6}} \\ \end{pmatrix}$$
Indeed, we can verify that the matrix is orthogonal and $Q \, (1, 1,1)'=(\sqrt{3} ,0, 0)'$
Lets consider first the upper part. That corresponds to the points spanned by the edges starting on the upper vertex, hence they correspond to the revolution of:
$$ \begin{pmatrix} x \\y \\ z \end{pmatrix} = Q \begin{pmatrix} 1 \\\alpha \\ 1 \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{3}}(\alpha+2) \\ \frac{1}{\sqrt{2}}(\alpha1)\\ \frac{1}{\sqrt{6}}(\alpha1) \end{pmatrix} $$
Then $\alpha=\sqrt{3} x2$, in the range $\alpha \in[1,1]$, or $x\in[1/\sqrt{3},\sqrt{3}]$.
The rotations along the $x$ axis will keep $r^2=y^2+z^2$ constant, and so
$$ r^2= \frac{2}{3}(\alpha1)^2=\frac{2}{3}(3\sqrt{3}x)^2$$
Or $$r = \sqrt{6}\left(1\frac{x}{\sqrt{3}}\right)$$
For the next part, we consider another edge, starting from a neighbour vertex, say from $(1,1,1)'$:
$$ \begin{pmatrix} x \\y \\ z \end{pmatrix} = Q \begin{pmatrix} 1 \\\alpha \\ 1 \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{3}}\alpha \\ \frac{1}{\sqrt{2}}(\alpha+1)\\ \frac{1}{\sqrt{6}}(\alpha3) \end{pmatrix} $$ with $\alpha=\sqrt{3} x$, in the range $\alpha \in[1,1]$, or $x\in[1/\sqrt{3},1/\sqrt{3}]$.
And here the radius is
$$ r^2= y^2+z^2=\frac{2}{3}\alpha^2+2=2x^2+2$$
Then the radius is
$$ r(x)=\begin{cases} \sqrt{6}\left(1\frac{x}{\sqrt{3}}\right) & \mbox{if } 1/\sqrt{3} \le x \le \sqrt{3}\\ \sqrt{2x^2+2} & \mbox{if } 0 \le x \le \sqrt{1/3} \end{cases}$$
(This seems to agree with String's answer.)
So, yes the middle cross section is an hyperbola.
To compute the total volume you need to integrate : $V = 2 \int_0^\sqrt{3} \pi r(x)^2 dx $ and scale the result by multiplying it by $(L/2)^3$ (because our cube has edge length $2$, instead of $L$)
Solution 3
Hint:
This rotationsurface is called Katenoid.
It's parametrization is given by
$$x(u,v)=(a \cos (u) \cosh (v),a \sin (u) \cosh (v),a v)$$
To calculate the volume, use the function:
$$f(x)=a \cosh \left(\frac{xA}{a}\right)+B$$.
Rotate it around the xAxis.
Then a formulae for volume is given by:
$$V=\pi \int _a^b\ f(x)^2{dx}$$
Please, look at my comment again. Here is another rotationsurface rotationhyperboloid.
$$x(\text{s},\text{v})\text{=}(\cos (s)v \sin (s),v \cos (s)+\sin (s),v)$$
This one can be generated by straightlines:
I don't know, which surface to use.
Solution 4
The central portions are hyperboloids of 1 sheet capped by two cones at the ends. ( not catenoids). The hyperboloid of one sheet is a ruled surface formed by rotation of a skew line about the rotational axis of symmetry.(Examples are cooling towers, hypoid gears etc).
Equations
Hyperboloid of one single sheet $ \dfrac{(x^2 + y^2)}{a^2} \dfrac{z^2}{c^2}=1 $ with one negative sign,
Hyperboloid of two separate sheets$ \dfrac{(x^2 + y^2)}{a^2} +\dfrac{z^2}{c^2}=1 $ with two negative signs.
Volume calculation is done usual way of integration using meridian curve equation given above.
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Vim
Updated on March 18, 2020Comments

Vim about 3 years
We have a cube with edge length $L$, now rotate it around its major diagonal (a complete turn, that is to say, the angle is 360 degrees), which object are we gonna get?
Astoundingly the answer is D. And here is a demonstration:
Well now I'm required to calculate the volume of this monster. It's far beyond my capability. I don't know how to analytically describe the curve between the two cones (although I believe it is a hyperbola). And I'm still not sure why it should be a curve rather than a segment or something. Could you help me? Thanks in advance.
Vim about 8 yearsThis is a problem in my 3d analytical geometry test and really drives me mad. I been thinking on it for a week and still can't even interpret the demo. ðŸ˜”

krirkrirk about 8 yearsIs the angle of the rotation given ?

Vim about 8 years@krirkrirk of course it's a whole rev. (360 deg)

Vim about 8 years@krirkrirk. Are my two pictures properly displayed? Perhaps there are some problems with picture display on this page. But on my cellphone it is ok.

krirkrirk about 8 yearsIt's perfectly clear, good job :)

Andrew D. Hwang about 8 years(+1) If you have reason to edit the question in the future, you might mention explicitly that the solid in question is swept out, i.e., is the union of all positions of the cube during one full turn. As is, the wording could be interpreted to mean the solid is the intersection of all the cube's positions, which would be a convex solid. (This point isn't worth a special edit, however.)

Vim about 8 years@user86418 thank u for good advice. Well I guess it'd be even harder in the "intersection" case than this one ;)

String about 8 years@Vim: For the intersection case, it becomes surprisingly much simpler, since then the curve is composed by the lines $f(x)=x/\sqrt2$ and $g(x)=(x\sqrt3)/\sqrt2$ which if desired can be combined to $$h(x)=(\sqrt6\sqrt 22x\sqrt3)/4$$ in which case the solid of revolution is simply composed of two cones.

robjohn about 8 yearsThis seems to be a duplicate of this question.

String about 8 years@robjohn: Nice answer and animation you have there!

Vim about 8 years@String. Sorry but I don't understand how to determine the curve for the intersection case. Seems like it can't be done the same way as the union case is.

String about 8 years@Vim: I added that case to my animated diagram in my answer. Have a look and let me know if that helps. Edges describe paths furthest away from a given diagonal of the cube, but a diagonal of a side describes a path closest to a diagonal of the cube.


Vim about 8 yearsThank you a lot. But could you briefly explain why the surface should be a Catenoid if it isn't too deep? I just don't know how to analytically start.

Frieder about 8 yearsThe picture (D) is a good advise for me. The edges of the cube (the longer ones) are generating cones. The smaller ones "vanish" inside. Seem's very regular to me. Do Carmo may help (Differentialgeometry) Chapter 4.2

Frieder about 8 yearsAs far as I know, one can generate the Catenoid also with straightlines in $\mathbb{R}^3$. But for me, it's far in the past. Now geometry is not subject, but I love it.

Vim about 8 yearsWell differential geometry is far ahead of me since I'm still learning elementary math analysis.. So at least one thing I'm sure is that this problem is far too difficult to appear in that test, then it should be my prof's fault rather than mine that I couldn't solve it :). And thank you again for all your time

Frieder about 8 yearsI'm with you! It is too difficult within a test.

Vim about 8 years.......but the other answer says it's a Catenoid. Do you think that answer is wrong?

String about 8 years@Vim: I will think about it a little more and eventually return. I just plugged the setup into my knowledge of 3Dgeometry and this popped out.

String about 8 years@Vim: It appears several people agree that this should be the curve, not the catenoid. How do you like my 3Dmodel :)

Steven Stadnicki about 8 yearsThis answer is just wrong  the catenoid is not a ruled surface. The correct answer is that it's a section of a hyperboloid (specifically, a hyperboloid of one sheet).

abel about 8 yearsstring, really neat. i just made a paper model. i am cutting the petrie polygon into half. i have a hexagon and equilateral triangle for the top and bottom. i need to figure out how to parametric the intermediate irregular hexagons.

abel about 8 yearsnice. it will take sometime for me to fully understand. will try.

Frieder about 8 years@Steven: My first view on picture (D) above made me thought, that could be a catenoid. Now it's very clear. Great work done! Thank you all together.

Frieder about 8 years@String: Excellent animation!

Andrew D. Hwang about 8 years@String: (+1) It might be easier to calculate the orthogonal distance from an edge to the axis using linear algebra rather than minimizing a quadratic? Anyway, nice solution. :)

String about 8 years@user86418: Thank you! Yes, that might be easier  although pojections are also tedious when working with variable coordinates. I have not considered that very carefully, though.

String about 8 years@Frieder: I am glad to hear you like it! I enjoyed adding that animation part very much myself. It took some extra work! Animations in GeoGebra has become kind of "my thing", as can be seen several places in my answering history.

Frieder about 8 years@String: I like to build animations with mathematica. I didn't know much about GeoBra, just that it exists. And once more: Very nice!

Vim about 8 yearsBrilliant demo! @String

Vim about 8 yearsI'm sorry that it was too late yesterday and I went to bed. I'm just getting up and will take a close look at your post. Thank you so much!@String

Vim about 8 years@String incredible! I believe that's the best answer a freshman can understand.

Vim about 8 years1 million thanks!