Question 861272: My art students and I have an applied math question in the form of a word problem:
My art students and I are creating mosaics on 7 cylindrical concrete planters located at the front entrance of our school. The cylinder dimensions are H=24" and D=34.7" (OK to round up to 36"). We cannot imagine making these mosaics with 1" tiles, which would mean hand placing over 18,000 tiles on approximately 130 square feet of surface area, in the next two months! What is the largest tile width we can lay on these large cylinders which will appear to fit the surface of the cylinder. We are likely using glass tile and stained glass which are 1/8" thick. The tiles will be glued using Thinset mortar at about 1/16" thick. Once all tiles are in place, the mosaics will be grouted to weatherproof, fill gaps and unify the entire "look" of the image.
I recently spoke with a world-renowned mosaicist who created a similar-sized cylindrical mosaic. She told me she believes she was able to use tiles about 5-6" wide without having noticeable sharp edges sticking out.
I really appreciate your help!
Sincerely,
Julie Muellejans
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! DISCLAIMER: I am not an artist. I am just a math geek, so I can give you some calculations, but
You are right about the work involved in using 1" tiles.
With a diameter of 34.7", the perimeter of one planter (in inches) would be
(34.7)(3.14) = about 109
Then, the surface area for just one planter (in square inches) would be
(24)(109) = 2616, and for all 7 planters it would be over 18,000 square inches.
However, a 2" tile has a surface area of 4 square inches,
and wit 2" tiles the number of tiles would be about 18,000/4 = 4,500.
Better yet, with 3" tiles (each one covering 9 square inches),
you would use about 18,000/9 = 2,000 tiles.
With 4" tiles, you would use about 1,100,
and with 6" tiles, you would need about 500.
On the other hand, avoiding sharp angles and gaps between planter and tile could limit the size of tile you can use. Besides, with 500 tiles (about 70 tiles per planter) your mosaic design may not be too interesting.
Calculations are fine, but I would try just one circle of tile (maybe temporarily/loosely attached) around the perimeter of a planter (or a cylinder with the dimensions of the planter), to see how it fits and looks.
Dividing the circumference of the planter by the tile size, gives a good estimate of how many tiles fit in one circle around the planter.
From a circumference of 109 inches, I found:
6" tiles --> 18.17 tiles (so 18 would fit)
5" tiles --> 21.8 tiles (refined calculations suggest that 22 may fit incredibly tight)
4" tiles --> 27.25 tiles
3" tiles --> 36.33 tiles
2" tiles --> 54.5 tiles.
The number of tiles determines the angle between the tiles.
With 6" tiles, you get 18 tiles per row around the circumference of the planter, and the cross section of the tiled planter would be an 18-sided polygon.
As you go around an 18-sided polygon, at each corner you change direction by an angle of
Edge on, the side of the planter would look like this:
The drawing (to scale) represents a 17.41" radius circle (37.4" diameter planter, covered by a 0.06" layer of glue), with 1/8" thick 6" tiles glued around it.
The tiles will be glued at their centers, but the edges will be about 0.27" or 1/4" from the curved surface of the planter. Would that gap be a problem?
The angle between tiles would be  . Is that a problem?
Each planter would have 24/6 = 4 rows of 18 tiles, for a total of 72 tiles. Is that too few tiles for your design?
If any of those issues is a problem, smaller tiles would lessen the problem.
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