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Topic: how many triangles can a human distinguish (Read 1821 times) |
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TheoH54
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how many triangles can a human distinguish
« on: May 24th, 2012, 7:16am » |
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Suppose the human eye can distinguish a difference of one degree between two angles (which is about right), how many different triangles are there to the human eye? In other words: how many different triangles are there with angles an integer number of degrees ? Of course ignoring flipping, rotation & resizing. 
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towr
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Re: how many triangles can a human distinguish
« Reply #1 on: May 24th, 2012, 11:10am » |
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It's just a matter of counting the triples, a,b,c such that 1<=a<=b<=c<=178 and a+b+c=180
So, I think:
sum(c=60..178, (179-c)-ceil((180-c)/2) + 1 ) = 3600
[edit]ah, wait, I forgot b <=c, so sum(c=60..89, c-ceil((180-c)/2) + 1 )+sum(c=90..178, (179-c)-ceil((180-c)/2) + 1 ) = 2700 [/edit]
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| « Last Edit: May 24th, 2012, 11:20am by towr » |
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Wikipedia, Google, Mathworld, Integer sequence DB
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pex
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Re: how many triangles can a human distinguish
« Reply #2 on: May 24th, 2012, 11:53am » |
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| hidden: | The number of triples of positive integers (a,b,c) with a+b+c=180 is (180-1 choose 2-1) = 15931. In this total:
- one triangle, (60,60,60), has been counted once;
- 88 triangles, (1,1,178) ... (89,89,2) except (60,60,60), have been counted three times;
- all other triangles have been counted 3! = 6 times.
Thus, the number of triangles is the solution to 1*1 + 88*3 + (n-1-88)*6 = 15931, which matches towr's n = 2700. |
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