What is the largest volume object that can pass though a $1 \times 1 \times L$ "snaky" corridor, where $L$ is large enough to be irrelvant, say $L > 6$.


This is a 3D version of the 2D sofa-moving problem, which has been heavily studied. See especially Dan Romik's web pages. The optimal-area 2D sofa is conjectured to be Gerver's (slight) modification of Hammersley's shape, the latter of which I show below, extruded in 3D to fill the corridor.

There are two natural candidates: (1) Slice the extruded 2D optimal shape, in the orthogonal direction, so it can negotiate both turns in the same manner. See image added below. (2) Rotate the illustrated shape $90^\circ$ but shear-off every portion that falls outside the $1 \times 1$ corridor.

A basic question is: Is either of these the optimal solution, or can one identify some shape that beats both? An even more basic (and easier) question is: Which of (1) or (2) has larger volume?


          The intersection of the two shapes can pass through the corridor.
          The intersection. (Thanks to J.M. & JackLaVigne @MathematicaSE.)
Volume: $\frac{4 \left(8+\pi^3\right)}{3 \pi ^3} \approx 1.67735$.

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    $\begingroup$ Maybe there is a way to make a simpler 3D-analog of the Sofa problem, with just one turning point, by requiring that one of the solid's axes must be aligned in some direction before and in some other direction after ? $\endgroup$ – F. C. Aug 6 '16 at 10:12
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    $\begingroup$ To rephrase my previous comment, contrary to the plane case, the shape can arrive in its final position in 4 different ways. This gives 4 distinct problems. $\endgroup$ – F. C. Aug 6 '16 at 20:00
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    $\begingroup$ Also, what kind of building are you moving into? $\endgroup$ – Simon Rose Aug 16 '16 at 8:35
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    $\begingroup$ The shape of egg may be connected with this problem. $\endgroup$ – Takahiro Waki Mar 23 '17 at 14:23
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    $\begingroup$ Another natural 3D variant of the question would be a pipe with circular rather than square cross-section and bends allowed in arbitrary direction. One can also ask questions regarding what the maximality of the body's volume implies about the body's symmetries: 1. For the pipe with square cross section and 4 bend direction must the maximal body have D_8 symmetry? 2. For the pipe with circular cross-section and arbitrary bend direction must the body have O(2) symmetry? $\endgroup$ – Michael Feb 22 at 16:42

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