## June 18, 2010

### Light Rings

The ceiling light fixtures in the Texas Ballroom at the Hyatt Regency Austin have an interesting geometry. Each one consists of four concentric cylinders.

The configuration suggests a variety of mathematical questions concerning, say, relative area or volume.

Even more intriguing, however, are the rings of lights between the cylinders. One light sits in the center circle, eight in the first ring, ten in the second ring, and sixteen in the third. Is there a pattern?

We have the sequence: 1, 8, 10, 16, which suggests the following question: If the fixture were to have a fourth ring, how many lights would it contain?

Neil Sloane's On-:Line Encyclopedia of Integer Sequences is one source of possible answers. Querying the database, however, produces just one result for the string 1, 8, 10, 16. These are four consecutive numbers representing the absolute difference between the sum of the odd digits and the sum of the even digits of the nth prime, starting at n = 19. The next number in this sequence is 5, and that doesn't make sense in the context of the light fixtures.

So, is there a pattern, beyond the notion that the number of lights should increase as you move away from the center? What would be a plausible number for the fourth ring?

Photos by I. Peterson

hahahaha those things make me smile.

The world is a beautiful place

Anonymous said...

Also see the sectioning lines in the cylinders, which follow the pattern 4,8,12,12. This sequence produces more suggestions for the next ring, including 8,12,16,21,24... Assuming this is an increasing sequence I would guess the next cylinder would have 16 sections. There seems to be no pattern between this and the number of lights, but to make their averages equal, the next ring of lights would have roughly 17 lights. It would be awkward for an even number to appear in this sequence, so perhaps either 16 or 18 would be appropriate as the next number.

Also, it appears that the next cylinder would not be much bigger in diameter, so it might need more lights. Hence it is reasonable to assume that the next ring of lights has more than 16 of them. From the above points, my guess is that 18 is the most probable continuation.

Pacha Nambi said...

The second figure reminds me of the way DNA is packaged inside DNA toroids - where there is hexagonal packing.