Doron Posted October 1, 2003 Report Share Posted October 1, 2003 Hi, Please look at the attached pdf (123.pdf.pdf) You will find in it a circle and some sub-area inside it. The sub-area exists between the radius and a curve. The curve is connected to both sides of some radius, and goes through the intersection points that exists between n radii and n-1 inner circles, where each radius is divided by the inner circles to n equal parts. I have found that the sub-area(magenta) = circle's-area/3(cyan) 1) Can someone show why the magenta area = 1/3 of the cyan area ? 2) We can take any number of inner radii-circles intersection points, and create some border, which is made of straight lines between these points. By doing this, we get a closed polygon (an area). Now we take some closed polygon, find the total number of its vertexes and omit 2 (tolal - 2 = n). By doing this, we get some Natural number n which is conncted to some polygon area S (please see the attached pdf named Natural-areas.pdf.pdf). Through this way we can put in 1-1 correspondence some n with some S. When have this map, we can ask: S1 is the area of some polygon, where the number of totel-2=aleph0. S2 is the area of some polygon, where the number of totel-2=2^aleph0. 3) Is S1 = S2 ? 4) If the answer to (3) is no, then what is the difference between the two magenta areas, and how this difference related to the CH problem ? 5) Do you think that we have here some useful mathematical constant ? Thank you. Doron Link to comment Share on other sites More sharing options...
Tormod Posted October 1, 2003 Report Share Posted October 1, 2003 Hello Doron, and warm welcomes to our forums! I am sorry that I don't have the time to ponder your questions right now - they are very interesting - but I'm going away for a few days. I hope other members take up the challenge! Until later,Tormod Link to comment Share on other sites More sharing options...
deamonstar Posted October 2, 2003 Report Share Posted October 2, 2003 the first pdf is a diagram of a curve corresponding to PHI (also referred to as "the golden ratio" it is also found within "the golden triangle"). it is interesting to note that nature has uses this for many various stuctures from tree branching patterns to chonch shells all the way up to the structures of galaxies. for more on this go here http://www.intent.com/sg/ or here http://members.aol.com/lizanya/sound.html your second pdf appears to be deriving points from PHI in a fractal pattern. here is a page with other very informative links with diagrams and formulae. http://ccins.camosun.bc.ca/~jbritton/jbfunpatt.htm#TOPIC3 Link to comment Share on other sites More sharing options...
Doron Posted October 4, 2003 Author Report Share Posted October 4, 2003 Hi deamonstar, Thank you for your reply. I have found that this curve is the Archimedean sprial. A detailed information on it can be found here: http://www.scit.wlv.ac.uk/university/scit/modules/mm2217/int.htm Thank you for the websites addresses. Doron Link to comment Share on other sites More sharing options...
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