It all started last week, when Acme Engineering loaned him a prototype version of their 3D Home Printer (this was as a reward for Ernie providing them, patent-free, with a suitable building material - fast-setting foam-expanded ametriohylosambranic frothcrete). Ernie's always wants to explore the limits of a new tool, but the printer was a very early beta-version, so the results did not quite match normal architectural expectations. After a couple of hours of 'testing', Ernie's house had sprouted two domes, a small minaret, a three-story keep (complete with portcullis gate) and, at the back - a jumble of oddly shaped, stacked, and overlapping rooms that were supposed to become his new laboratory. To make things worse, frothcrete is an eye-wateringly off-putting shade of mustard and the lab-rooms had been printed without any electrical or plumbing fixtures. "It's a disaster!" said Ernie, "I have to paint minaret, fresco the domes, wire up the labs and install a bunch of plumbing. I just don't have time for all these jobs!".

Now I am terrible on ladders, so couldn't help with the painting or frescoing, and don't know the first thing about electricity, so couldn't contribute to the wiring. But I thought I might be able to do something with regards to the plumbing. Via a distracted conversation while Ernie was mixing paints and stirring plaster, I gleaned that there were six lab rooms that needed plumbing connections as follows: an oil pipe between the attic lab and the basement lab; a water pipe between the front lab and the back lab; and a liquid-helium pipe between the west lab and the east lab. The labs all adjoined a 1x1x1 m 'void' space, and the plumbing could all be routed through that.

I am not handy with pipe-cutters or wrenches, so didn't think it would be a good idea to actually install anything. But surely, I could help with planning? As I wasn't quite sure how big each pipe needed to be I decided to make them all as large as possible. I spent quite a bit of time drawing up a careful diagram (complete with colour-coding), in which the pipe gracefully twisted around each other as they crossed the 1m^3 void space, and left it on Erne's desk. I then headed off to an afternoon appointment - confident that Ernie would be pleased by my efforts when he saw my sketch. enter image description here

So, I am sure you can understand my disappointment when I dropped later in the evening and saw that my efforts had been somewhat rigorously dismissed with the aid of a large rubber stamp. To be honest, I felt more than a little hurt that my labours had been spurned so dismissively, and I stormed off in a huff. I can only guess how Ernie realised he had been a bit brusque, (quite possibly his wife heard about what happened and left his "Social Skills for Beginners" book on his bed-side table), because the next morning he popped over with a very nice half case of Gewurztraminer and offered the nearest thing to an apology that he can manage. "Maybe I'm a tiny bit to blame", he explained, "because I didn't clarify that the 1x1x1 m cube wasn't actually empty, but was instead filled with frothcrete, and that I wanted conduits cut through it, so I could add whatever pipes and cables I needed". Then he explained that he often used the rubber stamp to reject his own ideas if they weren't up to scratch "So you shouldn't take it to heart so much".

I immediately forgave him, and asked if there was any job that he needed done - so long as he could explain clearly and precisely what he needed. He then gave me a task as follows:

  • In Ernie’s house there are a set of twelve laboratory rooms separated by a frothcrete void space in the shape of a regular dodecahedron with edge length of precisely 1 m.

  • Ernie would like six conduits to be cut - one between each of the six pairs of parallel faces of the dodecahedron.

  • Each conduit should be of constant cross-section along its entire length.

  • All the conduits should have identical cross-section shape, with the largest possible cross-section area.

  • No two conduits should intersect with each other, or with the surface of the dodecahedron except at the associated entrance and exit faces.

PS: Ernie is now a little concerned that I haven't come up with an optimal solution yet. I told him that I was still having difficulty understanding some of the specifications so he told me:

  • Any length is fine, so long as it gets between the two sides of the dodecahedron.

  • Ernie judges a conduit to be of constant cross-section, with cross-section area of pi*r^2, if a sphere of radius R can be pushed through the entire length of the conduit, but couldn't be pushed through if the conduit was any smaller at any point.

I think this is a great chance to assist Ernie (and maybe get the other half case of wine as a reward), but I'm having difficulty even picturing a dodecahedron in my head - let alone working out what shape, path, or cross-section for the six conduits is going to be optimal. Can anyone assist?

  • $\begingroup$ is misspelling "having" part of the puzzle? I'm having an "overanalyze everything" moment here $\endgroup$
    – Someone
    Commented Apr 22 at 0:35
  • 2
    $\begingroup$ Sometmes my fngers are havng dfficulty httng all the correct keys. Now edted correctly thnk. $\endgroup$
    – Penguino
    Commented Apr 22 at 1:12
  • 1
    $\begingroup$ @Someone ...and nice to know that at least Someone apears to have read the whole thing... $\endgroup$
    – Penguino
    Commented Apr 22 at 1:15
  • 1
    $\begingroup$ How exactly is the shape of the cross-section defined, for non-straight conduits? $\endgroup$
    – Gareth McCaughan
    Commented Apr 22 at 1:17
  • $\begingroup$ I am not sure if you will need anything as complicated as this, but I imagine Ernie would probably suggest the following. For any conduit C, and any point p that lies within C, consider the set of all planar open surfaces {A} such that a is an element of {A} if and only if p lies on a and a is entirely bounded by the surface of C. The 'local' cross-section LCS of C at the point p is the member of {A} with the smallest cross-section area. If for some p in C there are two or more candidates for LCS with different shapes, then C is not a valid conduit. $\endgroup$
    – Penguino
    Commented Apr 22 at 3:07


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