problem arranging different objects in a 'perfect' circumference

[Mark Beever]

I need to arrange 14 objects in a circle (around a circumference). Imagine a circular shaft built of bricks, with the long sides of each brick forming a circular wall, each brick will touch the next one at the corners of it's inside edge (like a perfect circlular necklace of bricks touching with as even angles as possible).

For kicks, I have bricks of different sizes, which makes this process more challenging.

There are three sizes of brick, intermingled in a special order (there is no repeat to the order). As this isn't a straight repeat of one object, and the objects are of a specific dimension how do I arrange them so that:

a) the three different sized bricks are in the order I want them over the 14 spaces which i require.

b) the perfect circumference is effectively retro-engineered to enable each of the three different lengths of brick touch at two corners only for each brick, balancing the perfect angles to achieve as perfect a circumference as possible.

Your help is greatly appreciated!

Mark

[islandmon]

Thinking in terms of chord segments & arc angles with 360? of closure required. Each random combination of the 3 sizes must result in closure based on the sum of the chords of the arc angles. The question is... are the arc angles describing the brick edge chords divisible by 2 or 3 or 5 or 7. The result will determine the final patterning based on the division of 360? = 2pi = 0.01745*360.

Therefore, each combination of chords forming each course of bricks must equal 2pi.

See math examples:

[islandmon]

On second thought

try these arc angles:

[ 0.1745r + 0.1571r + 0.1176r ] =0.4487r =2pi / 14

[Pat Stanford]

What you are really making is a 14 sided polygon with unequal sides. If you are to inscribe/circumscribe this on a circle, the circumference of the circle has to be equal to approximately the same as the total edge length of the sides. If this where a regular polygon (equal length sides), a 14 sided poly has 99.1% of the circumference compared to a circle.

My first try would be to add up the length of the sides, make a circle with a circumference equal to 99% of that length and start drawing the bricks.

Pat

[Charlie_P]

Further to the above...

Things could be simplified a little if you were building the rings of bricks into a well or column rather than having an array of single courses. In that case the circumference would have to be roughly similar for each course. Out of the 94 different combinations of the three bricks there are some (14 small bricks) which are much smaller than others (14 large bricks) which would make an interesting well. The circumferences of all these combinations forms a normal distribution (bell shaped curve) so presuming that you have a little tolerance and you wanted to maximise the variation for aesthetic reasons then you could pick out the central part of the distribution that gets you the most brick combinations for the least variation in circumference between courses.

You could for instance pick out the five most common circumferences which would give you approx 32 different brick combinations (depends on the brick sizes); working through the brick sizes would tell you how much circumference variation you could cope with an still have a column rather than a heap.

If however you've got a series of independent rings then revert to the above...

Edited April 15, 2008 by Charlie_P

[Mark Beever]

Thanks guys,

There are some really interesting thoughts there. Due to pressure of time I took a similar method to that suggested by Pat, and adjusted the circumference until the 'bricks' fitted together. I love challenges like this, and it's really interesting how many different solutions there can be to one problem.

Thanks for all your input, very much appreciated!

Mark