2
$\begingroup$

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?

$\endgroup$
9
  • $\begingroup$ 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

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.