Epsilon=One
09-30-2005, 12:24 PM
Brunardot Triangles
Brunardot Triangles are unending sequences of paired primitive Pythagorean triangles (www.CQthus.com/PTr) (not similar to a larger Pythagorean triangle, a² + b² = c², and all sides are integers) in which each triangle of the sequence can be mapped to the same Natural integer (www.101123.com/NI) that can be any Natural integer (www.101123.com/NI).
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One of the paired Brunardot Triangles has a hypotenuse, “h,” that is equal to the base, “b,” plus one; such as: 3-4-5 and 5-12-13 triangles. (h = b + 1)
The other paired Brunardot Triangle has a hypotenuse, “H” that is equal to the base, “B,” plus two; such as: 8-15-17 and 10-24-26 triangles. ( H = B + 2)
If “x” is any Natural integer (www.101123.com/NI) than the sides of the paired, primitive Pythagorean triangles (www.CQthus.com/PTr) are:a = 2x + 1
b = (a² – 1)/2
h = b + 1
A = a + 1
B = (A/2)² – 1
H = B + 2Amazingly, the generating Natural integer (www.101123.com/NI), “x,” is the radius of an inscribed circle that is common to each triangle of the Brunardot Triangle pair.
Radius of inscribed Pythagorean circle
x = (a + b – h)/2 = (A + B –H)/2.
All Brunardot Triangle pairs generate an acute and an obtuse Brunardot Ellipse (www.101123/BE) (BE) that each has a radius, “r,” equal to the altitude, “a,” (short leg) of its respective Brunardot Triangle. (Thus, it is possible to relate Brunardot Triangles to the Natural Emergent Ellipsoid (EEd), or fundamental quantum.
Examples of Brunardot Triangles
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http://i.g2d.us/x.gif
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Brunardot Triangles are unending sequences of paired primitive Pythagorean triangles (www.CQthus.com/PTr) (not similar to a larger Pythagorean triangle, a² + b² = c², and all sides are integers) in which each triangle of the sequence can be mapped to the same Natural integer (www.101123.com/NI) that can be any Natural integer (www.101123.com/NI).
(If no image appears below, "Click" your browser "Refresh" icon.)
http://i.g2d.us/bt2-350.gif
One of the paired Brunardot Triangles has a hypotenuse, “h,” that is equal to the base, “b,” plus one; such as: 3-4-5 and 5-12-13 triangles. (h = b + 1)
The other paired Brunardot Triangle has a hypotenuse, “H” that is equal to the base, “B,” plus two; such as: 8-15-17 and 10-24-26 triangles. ( H = B + 2)
If “x” is any Natural integer (www.101123.com/NI) than the sides of the paired, primitive Pythagorean triangles (www.CQthus.com/PTr) are:a = 2x + 1
b = (a² – 1)/2
h = b + 1
A = a + 1
B = (A/2)² – 1
H = B + 2Amazingly, the generating Natural integer (www.101123.com/NI), “x,” is the radius of an inscribed circle that is common to each triangle of the Brunardot Triangle pair.
Radius of inscribed Pythagorean circle
x = (a + b – h)/2 = (A + B –H)/2.
All Brunardot Triangle pairs generate an acute and an obtuse Brunardot Ellipse (www.101123/BE) (BE) that each has a radius, “r,” equal to the altitude, “a,” (short leg) of its respective Brunardot Triangle. (Thus, it is possible to relate Brunardot Triangles to the Natural Emergent Ellipsoid (EEd), or fundamental quantum.
Examples of Brunardot Triangles
(If no image appears below, "Click" your browser "Refresh" icon.)
http://i.g2d.us/x.gif
http://i.g2d.us/bt4-350.gifhttp://i.g2d.us/bt5-350.gif
http://i.g2d.us/bt6-350.gif
©Copyright 2005 by Brunardot. All rights reserved.
Terms: PhysicsMathForums.com, Brunardot, and Pulsoid Theory must be cited.
Sorry! This Thread has not been completed.
Please Bookmark and return to this site often.
If there is an immediate need for information,
please e-mail directly at the below "Click" link.
Every effort will be made to expedite a reply
with the requested information.Please ask questions. :)
With questions it’s possible to know if
comments are logical and convincing;
or whether clarification is required.
Please note that any private correspondence
may be edited and anonymously posted unless
requested otherwise.
If images are missing, "click" the Refresh Icon.
http://www.g2d.us/ws.gifhttp://www.g2d.us/s.gifhttp://www.g2d.us/t.gifhttp://www.g2d.us/m.jpghttp://www.g2d.us/i.gif
http://www.g2d.us/x.jpg