Scientific Approach to the Yin-Yang Geometry by Sergey Yu.
Shishkov (RUSSIA, Shishkovser@rambler.ru)
Here is given (below) the most generalized definition of the
astroid-like hypocycloid as the trajecory of a point P of a
rotating with angular velocity "omega1"=1 circle of radius
"radius1"=a, with centre of which also being rotating around the
origin by the circle of radius "radius2"=1-a , and angular
velocity "omega2"=-3, so that"radius1" +"radius2"=1, and
"omega2"/"omega1" =-3. Then for coordinates X[t], Y[t] of this
point P we have: X[t]=(a)*cos(t)+(1-a)*cos(3*t);
Y[t]=(a)*sin(t)-(1-a)*sin(3*t);
1-X[t]^2-Y[t]^2=factor(simplify(expand(1-((a)*cos(t)+(1-a)*cos(3
*t))^2-((a)*sin(t)-(1-a)*sin(3*t))^2)))=16*a*cos(t)^2*(cos(t)-1)
*(cos(t)+1)*(-1+a)=16*a*cos(t)^2*(cos(t)^2-1)*(-1+a)=16*a*cos(t)
^2*(sin(t)^2)*(1-a)=FULL SQUARE!=>
If Z[t]=4*cos(t)*sin(t)*(a*(1-a))^(1/2), then
X[t]^2+Y[t]^2+z[t]^2=1 ;(i.e., For every time t {X[t],Y[t],Z[t]} is on the unit SPHERE!!!).
With different values of the parameter a we obtain the
whole class of astroid-like hypocycloids with FOUR PARTS. Below
is given the Maple 5.4 Text programm for plotting of these
trajectories.;
> a=0.6339;plot([(a)*cos(t)+(1-a)*cos(3*t),(a)*sin(t)-(1-a)*sin(3*t)
,t=0..2*Pi]);
plot([(a)*cos(t)+(1-a)*cos(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2),
t=0..2*Pi]);
plot([(a)*sin(t)-(1-a)*sin(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2),
t=0..2*Pi]);
Look also in Maple 5:
> factor(simplify(expand(1-((b)*cos(t)+(1-b)*cos(3*t))^2-((b)*sin(
t)-(1-b)*sin(3*t))^2)));
> The Optimal Value for the parametr a is a=(1/2)*(3-3^(1/2))=0.6339, as will be
shown elsewher;-). It corresponds to the most "BEAUTIFUL" 3D-hypo-astroid. Such a configuration may serve also as the Yin-Yang
MAGNETIC TRAP for adiabatic freezing of Bose condensate in modern Atomic Beam Lasers and for hot plazma in thermonuclear fusion systems. However, this is beyond the scope of this site.
Let us call this unique value of the parameter a as "THE YIN-YANG PLATINUM SECTION", analogous to the famous "GOLDEN SECTION GS" (i.e.,GS=1/2*5^(1/2)-1/2=0.6180339887), suggested by Leonardo da Vinci!
>
[>a:=0.6339;plot([(a)*cos(t)+(1-a)*cos(3*t),(a)*sin(t)-(1-a)*sin
(3*t),t=0..2*Pi]);
plot([(a)*cos(t)+(1-a)*cos(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2),
t=0..2*Pi]);
plot([(a)*sin(t)-(1-a)*sin(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2),
t=0..2*Pi]);
Научный Подход к Yin-Yang Геометрии (Sergey Yu. Shishkov ,РОССИЯ, Shishkovser@rambler.ru) Здесь дается (ниже) наиболее обобщенное определение астроидно-подобной гипоциклоиды как траектории точки P вращающейся с угловой скоростью "omega1" =1 по окружности с радиусом "radius1" =a, центр которой также вращается вокруг начала координат по окружности с радиусом "radius2" =1-a, и угловой скоростью ""omega2" = -3, так, чтобы "radius1" + "radius2" =1, и "omega2" / ""omega1" = -3. Тогда для координат X [t], Y [t] этой точки P мы имеем: X [t] = (a) *cos (t) + (1-a) *cos (3*t); Y [t] = (a) *sin (t) - (1-a) *sin (3*t); 1-X[t]^2-Y[t]^2=factor(simplify(expand(1-((a)*cos(t)+(1-a)*cos(3 *t))^2-((a)*sin(t)-(1-a)*sin(3*t))^2)))=16*a*cos(t)^2*(cos(t)-1) *(cos(t)+1)*(-1+a)=16*a*cos(t)^2*(cos(t)^2-1)*(-1+a)=16*a*cos(t) ^2 * (sin(t) ^2) a* (1-a) =ПОЛНЫЙ КВАДРАТ! = >, если Z [t] =4*cos (t) *sin (t) * (a* (1-a)) ^ (1/2), то X [t] ^2+Y [t] ^2+Z [t] ^2=1; (то есть, Для каждого момента времени t {X [t], Y [t], Z [t]} С различными значениями параметра a мы получаем целый класс astroid-подобных гипоциклоид с ЧЕТЫРЬМЯ ЧАСТЯМИ. Ниже дается Maple5.4 Текст программы для составления графика этих траекторий.; > a=0.6339;plot([(a)*cos(t)+(1-a)*cos(3*t),(a)*sin(t)-(1-a)*sin(3*t), t=0 .. 2*Pi]); plot([(a)*cos(t)+(1-a)*cos(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2), t=0 .. 2*Pi]); plot([(a)*sin(t)-(1-a)*sin(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2), t=0 .. 2*Pi]); Просмотр также в МАПЛ 5: > factor(simplify(expand(1-((b)*cos(t)+(1-b)*cos(3*t))^2-((b)*sin (T) - (1-b) *sin (3*t)) ^2))); > Оптимальное Значение для параметра a = (1/2) * (3-3 ^ (1/2)) =0.6339, как будет показано в другом месте ; -). Этому значению соответствует наиболее "КРАСИВЫЙ" 3D-hypo-astroid. Такая конфигурация может служить также как Инь-Яньская МАГНИТНАЯ ЛОВУШКА для адиабатического охлаждения Бозе-конденсата в современных Лазерах на пучке атомов и для горячей плазмы в системах термоядерной реакции. Однако, это выходит за рамки данного сайта. Позвольте нам называть это единственное значение параметра как " ИНЬ-ЯНЬСКОЕ ПЛАТИНОВОЕ СЕЧЕНИЕ ", аналогично известному " ЗОЛОТОМУ СЕЧЕНИЮ З.С. " (то есть., З.С.=1/2*5 ^ (1/2) -1/2=0.6180339887), предложенному Леонардо да Винчи! > [> a:=0.6339;plot([(a)*cos(t)+(1-a)*cos(3*t),(a)*sin(t)-(1-a)*sin (3*t), t=0 .. 2*Pi]); plot([(a)*cos(t)+(1-a)*cos(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2), t=0 .. 2*Pi]); plot([(a)*sin(t)-(1-a)*sin(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2), t=0 .. 2*Pi]);
MY GUESTBOOK. Please, ANSWER THERE have you ever seen ball-lightning (Y/N), UFO and/or aliens and also ABOUT YOUR WEB-SITE if you have any!
CLICK HERE! THE ONLY ONE LINK TO OLD NEWMAIL!
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Where
does the Chinese Yin Yang symbol come from?
... Why
Yin Yang Symbol looks like this way? ... Find More Yin Yang
Sites.
Home Page Home of Chinese Fortune Calendar.
http://chinesefortunecalendar.com/yinyang.htm
-
YIN
& YANG AND THE I CHING
YIN 1 & YБNG AND THE I
CHING. ... Copyright (c) 1997 Kelley L. Ross, Ph.D.
All Rights
Reserved. Categories of Chinese Characters. ...
http://www.friesian.com/yinyang.htm
-
http://french.peopledaily.com.cn/beijing/yinyang.htm
La doctrine du yin et du yang. La doctrine du yin
et du
yang constitue une notion ...
http://french.peopledaily.com.cn/beijing/yinyang.htm
-
links
... Fritjof Capra Wendezeit
http://home.snafu.de/malzahn/Wissenschaft/Capra/Yin.htm,
Yin
und Yang http://www.tcminter.net/Artikel/yinyang.htm,
... Hafis und Rьckert, ...
http://www.yinyang-media.de/links.htm
-
Laura
Wulf: Light Years Photography
Laura Wulf, Light Years
Photography: women in sports, weddings and commitment ceremonies,
portraits, fine art
http://www.laurawulf.com/baseball/yinyang.htm
More Results From: http://google.yahoo.com/bin/query?p=YinYang.htm&hq=site:www.laurawulf.com
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Yin
Yuan Yang - ancient symbolism.
LINKS. YIN YANG.
http://chinesefortunecalendar.com/yinyang.htm.
http://chinesefortunecalendar.com/YinYang2.htm.
...
http://members.aol.com/yinyuanyang/links.htm
More Results From: members.aol.com
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Yin
Yang
... Yin Yang. Viele Kulturen haben eine
Energie gekannt, die der Bedeutung von ...
http://www.jacqueline-weichmann.de/yinyang.htm
More Results From: http://google.yahoo.com/bin/query?p=YinYang.htm&hq=site:www.jacqueline-weichmann.de
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Index
of /features
... Nov-2001 21:11 17k
person_year.htm 13-Nov-2001 21:11 14k tightrope.htm
11-Oct-2001
17:29 6k yinyang.htm 13-Nov-2001 21:11 12k
~temp00.htm 16-Sep-2001 21:19 18k
http://asiapacificuniverse.com/features/
More
Results From: asiapacificuniverse.com
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Yin-Yang,
by Laurie Green
... Yin-Yang in darr moo qigong
(http://tarming.sy.cei.gov.cn/qigong/yinyang.htm)
The
objective of regaining yin and yang balance for health is shown
to be the ...
http://faculty.wm.edu/jmchar/Kin493/kinst09b.html
-
List
of Files in "Learning the Tarot - An On-Line Course"
... Yin Yang Spread. yinyang.htm -
yinyang.jpg Charts. charts.htm chmaj.htm
chminat.htm
chmincc.htm chrkpk.htm
chrkqk.htm chst1.htm chst2.htm chcc.htm
...
http://www.learntarot.com/filelist.htm
More
Results From: http://google.yahoo.com/bin/query?p=YinYang.htm&hq=site:www.learntarot.com
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newyear
... edu/Angier/DimSum/Yin%26Yang%20Define.html 14.
http://www.americanwholehealth.com/library/acupuncture/yinyang.htm
15.
http://www.newton.mec.edu/Angier/DimSum ...
http://www.anthonyschool.org/CyberTreks/NewYear/newyear.htm
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Index
of /pagony/yinyang
... Parent Directory
06-Nov-2001 18:36 - hatter.jpg 02-Mar-1999 16:35
9k
yinyang.htm 15-Mar-2001 15:02 2k yinyang.jpg
02-Mar-1999 16:35 5k ...
http://www.akg.hu/pagony/yinyang/
More Results
From: http://google.yahoo.com/bin/query?p=YinYang.htm&hq=site:www.akg.hu
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Google
Directory - Society > Religion and Spirituality > ...
... Yin Yang Symbol -
http://chinesefortunecalendar.com/yinyang.htm An
explanation
of the Chinese Yin Yang Tai-Chi symbol using the I Ching.
...
http://directory.google.com/Top/Society/Religion_and_Spirituality/Divination/I_Ching/Research_Theori
es/
More Results From: directory.google.com
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List
of Files in "Learning the Tarot - An On-Line Course"
... Yin Yang Spread. yinyang.htm -
yinyang.jpg Summary Charts. charts.htm
chmaj.htm
chminat.htm chmincc.htm
chrkpk.htm chrkqk.htm chst1.htm chst2.htm
chcc.htm ...
http://www.wiccanway.net/tarot/filelist.htm
More Results From: http://google.yahoo.com/bin/query?p=YinYang.htm&hq=site:www.wiccanway.net
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YIN/YANG
... Chinese Glossary Tao Chinese Philosophy
Pre-Confucian
China and the Five Classics. ...
http://www.wsu.edu:8080/~dee/CHPHIL/YINYANG.HTM
More Results From: http://google.yahoo.com/bin/query?p=YinYang.htm&hq=site:www.wsu.edu
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http://srd.yahoo.com/goo/YinYang.htm/16/T=1010583232/F=4b2fe325e22a0dad0f8fa6259dd32a9b/*http://inst.augie.edu/~bastolz/paper-7.htm
... YIN & YANG AND THE I CHING.
http://www.friesian.com/yinyang.htm.
© 2000 Kelley L.
Ross, Ph.D. Date accessed: 4/29/2001. ...
http://inst.augie.edu/~bastolz/paper-7.htm
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Taoisme:
yin og yang
Taoismens grunnbegreper: Yin og yang. Yin og
yang
betyr egentlig den mшrke og den lyse siden ...
http://www.sv.uit.no/seksjon/religion/ansatte/hjemmesider/roald/kina_japan/yinyang.htm
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The
Feng Shui Fanzine
... 09. http://www.qi-whiz.com/.
10. http://www.brint.com/Systems.htm.
11.
http://members.aol.com/lordcolm4/us/yinyang.htm.
...
http://www.fengshui-fanzine.co.uk/Qi%20-%20A%20New%20Theory.htm
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Israel
Wheelers Team
... http://www.enc.qc.ca, Chile,
China,
http://www.chinesischer-staatscircus.de/CSC/YinYang.htm,
Colombia,
...
http://members.iol.co.il/alona/IsraelWheels/main_fram/links.htm
-
http://srd.yahoo.com/goo/YinYang.htm/20/T=1010583232/F=941ebcc9842742ca2f77ff027e10f9eb/*http://www.efn.org/~lorelei/news/jan2000.txt
... http://www.efn.org/~lorelei (Lorelei)
http://www.efn.org/~einhorn (Craig)
http://www.efn.org/~lorelei/yinyang.htm
(YinYang)
http://www.efn.org/~squeezbx (AA)
http://www.efn.org/~lorelei/news/jan2000.txt
More Results From: http://google.yahoo.com/bin/query?p=YinYang.htm&hq=site:www.efn.org
MATHEMATICS OF TAOISM