[math-fun] transcendental puzzles
A simple puzzle that leads to a transcendental equation: I'm going to cut a circular cookie into thirds (3 equal areas) by making two vertical cuts. Where should they be made? (I learned this from Gosper, but it must predate the invention of trigonometry. Probably just after cookies were invented.) A 3D version: The density of ice is .9, water is 1. A spherical iceberg of diameter 1 is floating in the ocean. How high is the tip of the iceberg over the surface of the water? Rich rcs@cs.arizona.edu
Hello, This problem reminds me of a geom. construction. Take 2 circles on the X axis of radius 1 , slide them one into each other until the common area is equal to the 2 others. Think as if you had a Venn diagram. When the 3 areas are equal then the height of the intersection is 0.739. The interesting thing about that is the number 0.739... it is the solution of the transcendental equation cos(x) = x. Funny isn't ? To get that number : go to my Inverter to get many digits or the OEIS to get some other infos : http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?An... But to get a <feeling> of what this number is : -take any scien. calculator like an HP something. -put the mode in RADIANS. -type 1. -Hit the COS button until it converges to the number. Unfortunately, I do not know other constructions that leads to other trans. equations too or the inverse. It is easy to generate trans. equations of course, what is more difficult is to invent a geom. construction to explain it. I had this hint about the geom. construction from Gilbert Labelle that read about it a while ago in the AMM (AMS monthly). Simon Plouffe
On Thursday, April 3, 2003, at 08:06 PM, Richard Schroeppel wrote:
A simple puzzle that leads to a transcendental equation: [...] The density of ice is .9, water is 1. A spherical iceberg of diameter 1 is floating in the ocean. How high is the tip of the iceberg over the surface of the water?
Do spheres have tips? (Now *that's* a transcendental question!) --Michael Kleber kleber@brandeis.edu
Reply-to: rwg@osots.com
I'm going to cut a circular cookie into thirds (3 equal areas) by making two vertical cuts. Where should they be made?
Of course, relaxing the "vertical" (or parallel) restriction buys nothing. In light of the seemingly contradictory identities cos(a) / [ 2 %pi I 2 sqrt(1 - x ) dx = --- - 2 a, ] 2 / sin(a) a >= 0 cos(a) / [ 2 %pi I 2 sqrt(1 - x ) dx = sin(2 a) + ---, ] 2 / - sin(a) we have in closed form the chords delimiting an area of pi/3 or anything we want. Unfortunately, the leftover "heel" segments have nondescript areas. Note that for area 2 pi/3, we need the peculiar quantity a = (asin pi/6)/2. Some transcendental equations at least have nice series solutions, e.g. y e^y = x -> inf ==== i - 1 i i + 1 \ (i + 1) (- 1) x y = > -------------------------- . / i! ==== i = 0 Can we similarly trisect a cookie with an infinite sum of a closed form? Or is this why Triscuits were square? --rwg
participants (4)
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Michael Kleber -
R. William Gosper -
Richard Schroeppel -
Simon Plouffe