宇宙的真实结构

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1 total messages Started by mavang2 Wed, 26 May 2004 12:06
#1397
Author: mavang2
Date: Wed, 26 May 2004 12:06
1583 lines
77356 bytes
���� ����ܡ� 
���������:��ȫ������ҳ֮��Ȩ�� 

������֮��ַ��: 
http://mavang2.tripod.com/square/index.html 


��ӭ���� mailto:zyeks@excite.com 
 

���ǻ����,��ҵ�����76��, 
�����72����76��Ͷ�����ɽ��ѧ���ϵ,У�����������,���������������Ժ��, 

����( ����Ĳ��𡢰���˹̹)˵�ķ�����������!ֻ���һ��������׼��Ҫ˵�Ĳ��Ǿ���Բ��! 
�ҽ�Ҫ��˵�������ʵ�ṹ, 
�ܾ���Բ�����һ�е���ѧ! 
�ҽ������ѧ����'squares theory'('��������'), 
���������к���'ͼʾ'��'����'������, 
��������֮�������binary(��Ԫ��), 
��Ԫ֮�����Ǳ˴�complementary��(�˴��෴��,�˴˲����), 
����,�޺�������������binary(��Ԫ��),�����Ǳ˴�complementary��(�˴��෴��,�˴˲����), 
����,��Ԫ֮���߱˴�'�ض�'�Ǵ���ƽ��ranking(��λ,���,��λ)! 
���,�޺����������߱˴�'�ض�'�Ǵ���ƽ��ranking(��λ,���,��λ)! 

squares theory�϶�˵: 
'һ������ƽ��!����������Ǹ�����֮�ֱ�����ֱ�! 
����������Ǹ�����ͬ����ƽ�ȵ������λranking!' 
'һ���޺�ƽ��!����޺����Ǹ��޺�֮�ֱ�����ֱ�! 
����޺����Ǹ��޺�ͬ����ƽ�ȵ��޺���λranking!' 

���ټ���һ��:binary֮�����Ǳ˴�complementary��,���Ǵ���ƽ��ranking! 

ÿһ��square��Ȼ���ı�,�ҹ涨ÿһ�ߵı߳�(֮���ֵ)һ����1! 
���ÿһ��square�����1, 
һ��square����������'С����'��ָ��һ���������ϵ 
���ض���Ƭ��֮'ʱ'(time)�� 
���ض�������֮'��'(site) 
��'��', 
�ҹ涨�úڰ���ɫ���������С����, 
'��'ֻ������,���Ǻڰ�����С����, 
��Ȼ,�����ַ�(��ڰ�����С����)֮�����Ǳ˴�complementary��, 
һ��square��֮С���ε�'����'��Ȼֻ������, 
����һ��square��֮�ڰ�С����֮'��Ŀ'����, 
��ɫС����֮����ܺ�һ����1/2, 
��ɫС����֮����ܺ�һ����1/2, 
����֮ 
'ʱ'��һ�������'��֮Ƭ�̺���֮Ƭ��֮�е�'�κ�Ƭ��֮ʱ, 
����֮ 
'��'��һ����ϵ��'��֮����Ͷ�֮����֮�е�'�κ�����֮��, 

square���õѿ������ϵ����ʾ��'ʱ'(time)��'��'(site), 
�ҹ涨 
square֮�����½Ǳ������ڵѿ������ϵ֮(0,0)��, 
square֮���±߱�����ѿ������ϵ֮+X axis �ص�,��+X axis��(0,0)����(1,0)��, 
square֮����߱�����ѿ������ϵ֮+Y axis �ص�,��+Y axis��(0,0)����(0,1)��, 
�ҹ涨 
+X axis��t axis,t��time,from 0 to 1, 
+Y axis��s axis,s��site,from 0 to 1, 

square�ı�֮����- 
�����:t=0,time����һ���������֮Ƭ��,s(site)����from 0 to 1,�˱���from(0,0)��to(0,1)��, 
���ұ�:t=1,time����һ���������֮Ƭ��,s(site)����from 0 to 1,�˱���from(1,0)��to(1,1)��, 
���±�:s=0,site����һ����ϵ�ĵ�֮����,t(time)����from 0 to 1,�˱���from(0,0)��to(1,0)��, 
���ϱ�:s=1,site����һ����ϵ�Ķ�֮����,t(time)����from 0 to 1,�˱���from(0,1)��to(1,1)��, 

ע��:s��t֮'���ֵ'����from 0 to 1,����'ʵ��ֵ'��ʵ��֮'������ٳ�'��'��ϵ�����'��, 

X_1 square֮ͼʾ:��[ͼʾx1] 
Y_1 square֮ͼʾ:��[ͼʾy1] 
[ͼʾx1]: 

http://mavang2.tripod.com/square/x1.bmp


Y-axis
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|________________________________________________________________|_X-axis
(0,0)                                                        (1,0) 
                                 

[ͼʾy1]: 


http://mavang2.tripod.com/square/y1.bmp



Y-axis
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|________________________________________________________________|_X-axis
(0,0)                                                        (1,0) 
                                 


The higher ranking formula: 
X_(n+1)=X_n.Y_n. ; 
Y_(n+1)=Y_n+X_n+ ; 
for n=1 to positive infinity ; 

��Ħ����: 
(.)'=(+) ; 
(+)'=(.) ; 

(X_n)'=(Y_n) ; 
(Y_n)'=(X_n) ; 

[X_(n+1)]'=[X_n.Y_n.]'= Y_n+X_n+ = Y_(n+1) ; 
[Y_(n+1)]'=[Y_n+X_n+]'= X_n.Y_n. = X_(n+1) ; 

����: 

X_2=X_1.Y_1. , 
Y_2=Y_1+X_1+ , 

X_3=X_2.Y_2. , 
Y_3=Y_2+X_2+ , 

X_4=X_3.Y_3. , 
Y_4=Y_3+X_3+ , 

�Դ�����,����,����recursive(�ݹ��,������,ѭ����)����, 
��Ԫ֮���߱˴�'�ض�'�Ǵ���ƽ��ranking(��λ,���,��λ), 
����X_2��Y_2�Ƕ�Ԫ֮����,�Ǳ˴�complementary��(�˴��෴��,�˴˲����), 
�˴�'�ض�'�Ǵ���ƽ��ranking(��λ,���,��λ), 
����(n+1)��n�Ƚ�ʱ, 
X_(n+1)��Y_(n+1)���ϼ�,higher ranking, 
X_n ��Y_n ���¼�,lower ranking, 
�ϼ����¼��ǲ�ƽ�ȵ�, 
�ϼ�,higher ranking�Ĺ�λ�Ƚϸ�, 
�¼�,lower ranking�Ĺ�λ�Ƚϵ�, 

������Ҫ���� 
{ 
X_2=X_1.Y_1. 
} 
֮����,���ȿ�X_2 square֮��������֮ͼʾ, 
�⼸������֮ͼʾ�������[ͼʾx2],[ͼʾx2-2],[ͼʾx2-3]��, 

 
[ͼʾx2]: 

http://mavang2.tripod.com/square/x2.bmp



Y-axis
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|(0,1)                          (1/2,1)                       (1,1)
|_________________________________________________________________ 
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|________________________________|________________________________|_X-axis
(0,0)                           (1/2,0)                       (1,0) 
                                 

[ͼʾx2-2]: 
http://mavang2.tripod.com/square/x2-2.bmp


Y-axis
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|(0,1)                          (1/2,1)                       (1,1)
|_________________________________________________________________ 
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|________________________________|________________________________|_X-axis
(0,0)                           (1/2,0)                       (1,0) 
                                 

[ͼʾx2-3]: 

http://mavang2.tripod.com/square/x2-3.bmp



Y-axis
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|(0,1)         (1/4,1)           (1/2,1)          (3/4,1)       (1,1)
|___________________________________________________________________ 
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|________________|________________|________________|________________|_X-axis
(0,0)         (1/4,0)           (1/2,0)          (3/4,0)        (1,0)
                                 



�������[ͼʾx2]��[ͼʾx2-2]��[ͼʾx2-3]��X_2 square֮��,�� 
��ɫ���ε�Ȼ������(ȫ�ڵ�)X_1 square, 
��ɫ���ε�Ȼ������(ȫ�׵�)Y_1 square, 
X_2 square��������������������: 
��һ,X_2 square֮��, 
ÿ����ɫ����,��time���ֲܷ��ں�ʱ,��site�ֲ������from 0 to 1����, 
ÿ����ɫ����,��time���ֲܷ��ں�ʱ,��site�ֲ������from 0 to 1����, 
�ڶ�,X_2 square֮��, 
��ɫ��������ܺ�һ����1/2, 
��ɫ��������ܺ�һ����1/2, 

ֻҪ�Ǻ�������������ص�square,�㶼�Ǵ�� 
{ 
X_2=X_1.Y_1. 
} 
֮X_2 square! 

[ͼʾx2]��[ͼʾx2-2]��[ͼʾx2-3]������X_2 square����ƽ�ȵ�X_2 ranking! 


������Ҫ���� 
{ 
Y_2=Y_1+X_1+ 
} 
֮����,���ȿ�Y_2 square֮��������֮ͼʾ, 
�⼸������֮ͼʾ�������[ͼʾy2],[ͼʾy2-2],[ͼʾy2-3]��, 
[ͼʾy2]: 
http://mavang2.tripod.com/square/y2.bmp



Y-axis
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|(0,1)                                                       (1,1)
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|________________________________________________________________|(1,1/2)
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|________________________________________________________________|_X-axis
(0,0)                                                        (1,0)
                                  

 

[ͼʾy2-2]: 
http://mavang2.tripod.com/square/y2-2.bmp



Y-axis
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|(0,1)                                                       (1,1)
|________________________________________________________________ 
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|                        white                                   | 
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|________________________________________________________________|(1,1/2)
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|________________________________________________________________|_X-axis
(0,0)                                                        (1,0) 
                                  


[ͼʾy2-3]: 
http://mavang2.tripod.com/square/y2-3.bmp



Y-axis
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|(0,1)                                                       (1,1)
|________________________________________________________________ 
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|                         black                                  |
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|________________________________________________________________|(1,3/4)
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|                       white                                    |  
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|________________________________________________________________|(1,1/2)
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|                       black                                    |
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|________________________________________________________________|(1,1/4)
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|                       white                                    |
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|________________________________________________________________|_X-axis
(0,0)                                                        (1,0)
                                  
�������[ͼʾy2]��[ͼʾy2-2]��[ͼʾy2-3]��Y_2 square֮��,�� 
��ɫ���ε�Ȼ������(ȫ�ڵ�)X_1 square, 
��ɫ���ε�Ȼ������(ȫ�׵�)Y_1 square, 
Y_2 square��������������������: 
��һ,Y_2 square֮��, 
ÿ����ɫ����,��site���ֲܷ��ںδ�,��time�ֲ������from 0 to 1����, 
ÿ����ɫ����,��site���ֲܷ��ںδ�,��time�ֲ������from 0 to 1����, 
�ڶ�,Y_2 square֮��, 
��ɫ��������ܺ�һ����1/2, 
��ɫ��������ܺ�һ����1/2, 

ֻҪ�Ǻ�������������ص�square,�㶼�Ǵ�� 
{ 
Y_2=Y_1+X_1+ 
} 
֮Y_2 square! 

[ͼʾy2]��[ͼʾy2-2]��[ͼʾy2-3]������Y_2 square����ƽ�ȵ�Y_2 ranking! 

������Ҫ˵��ο���square����֮��, 
��[ͼʾy2]Y_2 square������, 
���趨 
X_1 square��'����', 
Y_1 square��'����', 
�� 
Y_2 square֮�����ɫ���δ��: 
��һ������֮��,��һ����ϵ֮�°봦,�������'����'֮��, 
Y_2 square֮�����ɫ���δ��: 
��һ������֮��,��һ����ϵ֮�ϰ봦,�������'����'֮��, 
���˵�����з���𡢱���𡢻��������, 
Y_2 square���ĳһ�ַ���, 
����Y_2 square�Ƿ�Ϊ�����,�������, 

X_2 square֮ͼʾ:��[ͼʾx2], 
Y_2 square֮ͼʾ:��[ͼʾy2], 

[ͼʾx2]: 
http://mavang2.tripod.com/square/x2.bmp



Y-axis
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|(0,1)                          (1/2,1)                       (1,1)
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|________________________________|________________________________|_X-axis
(0,0)                           (1/2,0)                       (1,0) 
                                 

[ͼʾy2]: 
http://mavang2.tripod.com/square/y2.bmp


Y-axis
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|(0,1)                                                       (1,1)
|________________________________________________________________ 
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|                        black                                   | 
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|________________________________________________________________|(1,1/2)
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|                         white                                  | 
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|________________________________________________________________|_X-axis
(0,0)                                                        (1,0) 
                                  

 



X_3 square֮ͼʾ:��[ͼʾx3] 
Y_3 square֮ͼʾ:��[ͼʾy3] 


[ͼʾx3]: 
http://mavang2.tripod.com/square/x3.bmp



Y-axis
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|(0,1)          (1/4,1)          (1/2,1)                       (1,1)
|__________________________________________________________________ 
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|                |                |          black                 |
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|                |                |                                |
|                |                |________________________________|
| black          | white          |                         (1,1/2)| 
|                |                |                                | 
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|                |                |         white                  | 
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|________________|________________|________________________________|_X-axis
(0,0)                                                          (1,0) 
                                  
 

[ͼʾy3]: 
http://mavang2.tripod.com/square/y3.bmp



Y-axis
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|(0,1)                          (1/2,1)                       (1,1)
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|________________________________|________________________________|(1,1/2)
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|                       black                                     |
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|_________________________________________________________________|(1,1/4)
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|                       white                                     | 
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|_________________________________________________________________|_X-axis
(0,0)                                                         (1,0) 
    

����֮�ĸ�ͼʾ�����������ʾcomplementary������֮��ĸ�ϵ, 
��Ҫ��X_3 square[ͼʾx3]��Y_3 square[ͼʾy3]������, 
��Ȼ,X_3 square��Y_3 square�����Ǳ˴�complementary��, 

��һ. 
������X_3 square 
��[ͼʾx3] 

[ͼʾx3]: 
http://mavang2.tripod.com/square/x3.bmp

Y-axis
|
|
|(0,1)          (1/4,1)          (1/2,1)                       (1,1)
|__________________________________________________________________ 
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|                |                |                                |
|                |                |          black                 |
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|                |                |                                |
|                |                |                                |
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|                |                |                                |
|                |                |                                |
|                |                |________________________________|
| black          | white          |                         (1,1/2)|
|                |                |                                | 
|                |                |                                | 
|                |                |                                | 
|                |                |                                | 
|                |                |                                | 
|                |                |         white                  | 
|                |                |                                | 
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|                |                |                                | 
|                |                |                                | 
|                |                |                                | 
|                |                |                                | 
|                |                |                                | 
|________________|________________|________________________________|_X-axis
(0,0)                                                          (1,0) 
                         

�ڶ�. 
��X_3 square 
��ת����ת: 
��ת�Ƕ�: 
270��: 
��[ͼʾc-1] 

[ͼʾc-1]: 
http://mavang2.tripod.com/square/c-1.bmp
 


Y-axis
|
|
|(0,1)                          (1/2,1)                       (1,1)
|_________________________________________________________________ 
|                                |                                |
|                                |                                |
|                                |                                |  
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|                                |                                |  
|           black                |         white                  |
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|                                |                                |
|                                |                                |
|                                |                                |
|                                |                                |
|________________________________|________________________________|(1,1/2)
|                                                                 | 
|                                                                 |
|                                                                 |
|                                                                 |
|                       white                                     |
|                                                                 |
|                                                                 |  
|                                                                 |
|_________________________________________________________________|(1,1/4)
|                                                                 |
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|                                                                 |
|                       black                                     | 
|                                                                 |
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|_________________________________________________________________|_X-axis
(0,0)                                                         (1,0) 
                                  
  

����. 
��c-1 square 
��ת����ת: 
ˮƽ��ת: 
��[ͼʾc-2] 

[ͼʾc-2]: 
http://mavang2.tripod.com/square/c-2.bmp



Y-axis
|
|
|(0,1)                          (1/2,1)                       (1,1)
|_________________________________________________________________ 
|                                |                                |
|                                |                                |
|                                |                                |  
|                                |                                |
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|                                |                                |
|                                |                                |  
|           white                |           black                |
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|                                |                                |
|                                |                                |
|________________________________|________________________________|(1,1/2)
|                                                                 | 
|                                                                 |
|                                                                 |
|                                                                 |
|                       white                                     |
|                                                                 |
|                                                                 |  
|                                                                 |
|_________________________________________________________________|(1,1/4)
|                                                                 |
|                                                                 |
|                                                                 |
|                       black                                     | 
|                                                                 |
|                                                                 |
|                                                                 |
|_________________________________________________________________|_X-axis
(0,0)                                                         (1,0) 
                                  
  


����. 
��c-2 square 
ɫ�ʶԻ�: 
��[ͼʾc-3] 

[ͼʾc-3]: 
http://mavang2.tripod.com/square/c-3.bmp
 


Y-axis
|
|
|(0,1)                          (1/2,1)                       (1,1)
|_________________________________________________________________ 
|                                |                                |
|                                |                                |
|                                |                                |  
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|                                |                                |
|                                |                                |  
|           black                |         white                  |
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|                                |                                |
|                                |                                |
|                                |                                |
|                                |                                |
|                                |                                |
|________________________________|________________________________|(1,1/2)
|                                                                 | 
|                                                                 |
|                                                                 |
|                                                                 |
|                       black                                     |
|                                                                 |
|                                                                 |  
|                                                                 |
|_________________________________________________________________|(1,1/4)
|                                                                 |
|                                                                 |
|                                                                 |
|                       white                                     | 
|                                                                 |
|                                                                 |
|                                                                 |
|_________________________________________________________________|_X-axis
(0,0)                                                         (1,0) 
                                  
 



[ͼʾc-3]����Y_3 square! 

���� 
(X_3)'=(Y_3) , 
(Y_3)'=(X_3) , 
������֮�ĸ�ͼʾ����,����ʹ��һ.�����е�Y_3 square,�ڵ���.�������γ�X_3 square! 

�����: 
������X_n,�����Y_n, 
������Y_n,�����X_n, 

��֪:�޺�������������X_2��Y_2,�� 
X_2:����,���������������ĸо�, 
Y_2:�޺�,ŭ�������г�ª�ĸо�, 

X_3:ħ,���ֵ�ħ����, 
Y_3:��֧��,����ʹ��Ĺ?��, 

X_4:ת��ʥ��,'�ص���ȥ֮ʱ��'�ĸо�,����˼���������(��ο�:άĦڵ��˵��), 
Y_4:����,'�������¡�Ω�Ҷ���'�ĸо�, 


��֪:�޺�������������X_3��Y_3,�� 
X_3:����, 
Y_3:�޺�, 

X_4:ħ, 
Y_4:��֧��, 

X_5:����, 
Y_5:����, 

�޺����Ե�����Y_2, 
���Ǳ�֧�����Y_3, 

�޺����Ե�����Y_3, 
���Ǳ�֧�����Y_4, 

�޺����Ե�����Y_4, 
���Ǳ�֧�����Y_5, 

�Դ�����, 

X_n��Y_n���ɶԻ�,��X_n��(..)��ʽ,Y_n��(++)��ʽ, 
����ħ������ ����X_n��(..)��ʽ,����Y_n, 
�޺�����֧�𡢷��Ӷ���Y_n��(++)��ʽ,����X_n, 


���Ӳ����Ƿ����е��������! 
�ڷ���֮�ϻ��и��ranking(��λ,���,��λ)���ϼ�����,����������������, 
��������������'��������'��'����������', 
��������������:��������'����'����'������', 
��'¥���ƽ�'�ĸо�,��'������¥'�ĸо�, 
��Ȼ������������֮�ϻ��и��ranking(��λ,���,��λ)���ϼ�����, 

����������������:Խ���ϼ�,higher ranking,Խ����! 

������˵һ��square��, 
��ɫС����֮����ܺ�һ����1/2, 
��ɫС����֮����ܺ�һ����1/2, 
����X_1,Y_1�������ƺ��������, 
��ʵ��, 
X_1=X_0.Y_0. , 
Y_1=Y_0+X_0+ , 

X_0=X_-1.Y_-1. , 
Y_0=Y_-1+X_-1+ , 

X_-1=X_-2.Y_-2. , 
Y_-1=Y_-2+X_-2+ , 

�Դ�����,�������޴� ; 

X_0��Y_0�Լ����¼�,�������кڰ�����С����, 
����,����ɫ�������,���� 
�� 
X_-100 ��ȫ��ɫ, 
Y_-100 ��ȫ��ɫ, 
X_1��Ȼ��ȫ��ɫ, 
Y_1��Ȼ��ȫ��ɫ, 

��OK! 

-- 
[ͼʾx2],[ͼʾx2-2],[ͼʾx2-3]�����Դ��X_2,����ֻ��[ͼʾx2]�ܱ�����X_2֮'��������', 
[ͼʾx2-2]��[ͼʾx2-3]������X_2֮'��������', 
ÿ��X_nֻ��һ��'��������',ÿ��Y_nֻ��һ��'��������', 
for n=negative infinity to positive infinity. 
��Ӧ�ڵ�������,��Ҫ����'Z����'(Z function),Z����֮'������'(domain)��D, 
D={(x,y)|0С�ڵ���xС��1 and 0С�ڵ���yС��1 and x is real number and y is real
number.}
Z����֮'ֵ'(value)ֻ����-1/2��+1/2! 
��Z����֮��������D, 
��Z����û��ֵ,��0����x,��1С�ڵ���x,��0����y,��1С�ڵ���y, 

Z_X_n��һ��'Z����', 
If n=1,Z_X_n(x,y)=Z_X_1(x,y)=-1/2,for ALL points in D, 
Z_Y_n��һ��'Z����', 
If n=1,Z_Y_n(x,y)=Z_Y_1(x,y)=+1/2,for ALL points in D, 

��X_1 square��ȫ�ڵ�����,����Z_X_1(x,y)=-1/2,�� 
ÿ��X_n֮����������֮��ɫ����������֮ALL points֮Z����ֵ֮,����-1/2, 
ÿ��Y_n֮����������֮��ɫ����������֮ALL points֮Z����ֵ֮,����-1/2, 
for n=2 to positive infinity. 
��Y_1 square��ȫ�׵�����,����Z_Y_1(x,y)=+1/2,�� 
ÿ��X_n֮����������֮��ɫ����������֮ALL points֮Z����ֵ֮,����+1/2, 
ÿ��Y_n֮����������֮��ɫ����������֮ALL points֮Z����ֵ֮,����+1/2, 
for n=2 to positive infinity. 

��X_(n+1)=X_n.Y_n. 
�� Z_X_(n+1)(x,y)=Z_X_n(2x,y)+Z_Y_n(2x-1,y), 
If n=1 and 0С�ڵ���xС��1/2 and
0С�ڵ���yС��1,Z_X_(n+1)(x,y)=Z_X_n(2x,y)=Z_X_1(2x,y)= -1/2.
If n=1 and 1/2С�ڵ���xС��1 and
0С�ڵ���yС��1,Z_X_(n+1)(x,y)=Z_Y_n(2x-1,y)=Z_Y_1(2x-1,y)=+1/2.
��Y_(n+1)=Y_n+X_n+ 
�� Z_Y_(n+1)(x,y)=Z_Y_n(x,2y)+Z_X_n(x,2y-1), 
If n=1 and 0С�ڵ���yС��1/2 and
0С�ڵ���xС��1,Z_Y_(n+1)(x,y)=Z_Y_n(x,2y)=Z_Y_1(x,2y)= +1/2.
If n=1 and 1/2С�ڵ���yС��1 and
0С�ڵ���xС��1,Z_Y_(n+1)(x,y)=Z_X_n(x,2y-1)=Z_X_1(x,2y-1)=-1/2.

����, 
�� 
X_(n+2)=X_(n+1).Y_(n+1). 
�� 
Z_X_(n+2)(x,y)= 
Z_X_(n+1)(2x,y)+Z_Y_(n+1)(2x-1,y)= 
[Z_X_n(4x,y)+Z_Y_n(4x-1,y)]+[Z_Y_n(2x-1,2y)+Z_X_n(2x-1,2y-1)], 

��n=1,X_(n+2)=X_3,[ͼʾx3]��X_3֮��������, 
(1). 
If 0С�ڵ���xС��1/4 and 0С�ڵ���yС��1,then 0С�ڵ���4xС��1 and 0С�ڵ���yС��1, 
Z_X_3(x,y)=Z_X_(n+2)(x,y)=Z_X_n(4x,y)=Z_X_1(4x,y)=-1/2. 
��[ͼʾx3]��, 
{(x,y)|0С�ڵ���xС��1/4 and 0С�ڵ���yС��1} 
�Ǵ���߿�ʼ�����һ����ɫ��������, 
�������Ǻ�ɫ��������,�ʴ�������֮ALL points֮Z����ֵ֮,����-1/2, 
(2). 
If 1/4С�ڵ���xС��1/2 and 0С�ڵ���yС��1,then 0С�ڵ���4x-1С��1 and 0С�ڵ���yС��1, 
Z_X_3(x,y)=Z_X_(n+2)(x,y)=Z_Y_n(4x-1,y)=Z_Y_1(4x-1,y)=+1/2. 
��[ͼʾx3]��, 
{(x,y)|1/4С�ڵ���xС��1/2 and 0С�ڵ���yС��1} 
�Ǵ���߿�ʼ�����һ����ɫ��������, 
�������ǰ�ɫ��������,�ʴ�������֮ALL points֮Z����ֵ֮,����+1/2, 
(3). 
If 1/2С�ڵ���xС��1 and 0С�ڵ���yС��1/2,then 0С�ڵ���2x-1С��1 and 0С�ڵ���2yС��1, 
Z_X_3(x,y)=Z_X_(n+2)(x,y)=Z_Y_n(2x-1,2y)=Z_Y_1(2x-1,2y)=+1/2. 
��[ͼʾx3]��, 
{(x,y)|1/2С�ڵ���xС��1 and 0С�ڵ���yС��1/2} 
�Ǵ���߿�ʼ����ڶ�����ɫ��������, 
�������ǰ�ɫ��������,�ʴ�������֮ALL points֮Z����ֵ֮,����+1/2, 
(4). 
If 1/2С�ڵ���xС��1 and 1/2С�ڵ���yС��1,then 0С�ڵ���2x-1С��1 and 0С�ڵ���2y-1С��1, 
Z_X_3(x,y)=Z_X_(n+2)(x,y)=Z_X_n(2x-1,2y-1)=Z_X_1(2x-1,2y-1)=-1/2. 
��[ͼʾx3]��, 
{(x,y)|1/2С�ڵ���xС��1 and 1/2С�ڵ���yС��1} 
�Ǵ���߿�ʼ����ڶ�����ɫ��������, 
�������Ǻ�ɫ��������,�ʴ�������֮ALL points֮Z����ֵ֮,����-1/2, 


����, 
�� 
Y_(n+2)=Y_(n+1)+X_(n+1)+ 
�� 
Z_Y_(n+2)(x,y)= 
Z_Y_(n+1)(x,2y) +Z_X_(n+1)(x,2y-1)= 
[Z_Y_n(x,4y)+Z_X_n(x,4y-1)]+[Z_X_n(2x,2y-1)+Z_Y_n(2x-1,2y-1)], 

��n=1,Y_(n+2)=Y_3,[ͼʾy3]��Y_3֮��������, 
(1). 
If 0С�ڵ���yС��1/4 and 0С�ڵ���xС��1,then 0С�ڵ���4yС��1 and 0С�ڵ���xС��1, 
Z_Y_3(x,y)=Z_Y_(n+2)(x,y)=Z_Y_n(x,4y)=Z_Y_1(x,4y)=+1/2. 
��[ͼʾy3]��, 
{(x,y)|0С�ڵ���yС��1/4 and 0С�ڵ���xС��1} 
�Ǵ��±߿�ʼ�����һ����ɫ��������, 
�������ǰ�ɫ��������,�ʴ�������֮ALL points֮Z����ֵ֮,����+1/2, 
(2). 
If 1/4С�ڵ���yС��1/2 and 0С�ڵ���xС��1,then 0С�ڵ���4y-1С��1 and 0С�ڵ���xС��1, 
Z_Y_3(x,y)=Z_Y_(n+2)(x,y)=Z_X_n(x,4y-1)=Z_X_1(x,4y-1)=-1/2. 
��[ͼʾy3]��, 
{(x,y)|1/4С�ڵ���yС��1/2 and 0С�ڵ���xС��1} 
�Ǵ��±߿�ʼ�����һ����ɫ��������, 
�������Ǻ�ɫ��������,�ʴ�������֮ALL points֮Z����ֵ֮,����-1/2, 
(3). 
If 1/2С�ڵ���yС��1 and 0С�ڵ���xС��1/2,then 0С�ڵ���2y-1С��1 and 0С�ڵ���2xС��1, 
Z_Y_3(x,y)=Z_Y_(n+2)(x,y)=Z_X_n(2x,2y-1)=Z_X_1(2x,2y-1)=-1/2. 
��[ͼʾy3]��, 
{(x,y)|1/2С�ڵ���yС��1 and 0С�ڵ���xС��1/2} 
�Ǵ��±߿�ʼ����ڶ�����ɫ��������, 
�������Ǻ�ɫ��������,�ʴ�������֮ALL points֮Z����ֵ֮,����-1/2, 

(4). 
If 1/2С�ڵ���yС��1 and 1/2С�ڵ���xС��1,then 0С�ڵ���2y-1С��1 and 0С�ڵ���2x-1С��1, 
Z_Y_3(x,y)=Z_Y_(n+2)(x,y)=Z_Y_n(2x-1,2y-1)=Z_Y_1(2x-1,2y-1)=+1/2. 
��[ͼʾy3]��, 
{(x,y)|1/2С�ڵ���yС��1 and 1/2С�ڵ���xС��1} 
�Ǵ��±߿�ʼ����ڶ�����ɫ��������, 
�������ǰ�ɫ��������,�ʴ�������֮ALL points֮Z����ֵ֮,����+1/2, 

-- 

������˵������������Z����, 
������Ҫ˵�κ�һ��������Z����, 
��X_2��˵, 
[ͼʾx2],[ͼʾx2-2],[ͼʾx2-3]�����Դ��X_2,����ֻ��[ͼʾx2]�ܱ�����X_2֮'��������', 
��Ӧ��'��������'(typical square),���ǿ��Զ���'Z����',������˵: 
��X_(n+1)=X_n.Y_n. 
�� Z_X_(n+1)(x,y)=Z_X_n(2x,y)+Z_Y_n(2x-1,y), 
��Y_(n+1)=Y_n+X_n+ 
�� Z_Y_(n+1)(x,y)=Z_Y_n(x,2y)+Z_X_n(x,2y-1), 
���Ӧ�ڿ��Դ��X_2֮�κ�һ������,ǡ����һ��Z�������, 
��֪ 
Z_X_n(x,y)��X_n֮Z����,Z_Y_n(x,y)��Y_n֮Z����,(x,y)�ǵѿ������ϵ�ڵĵ�, 
Z����֮'������'(domain)��D, 
D={(x,y)|0С�ڵ���xС��1 and 0С�ڵ���yС��1 and x is real number and y is real
number.}
��n=1, 
Z_X_n(x,y)=Z_X_1(x,y)=-1/2,for ALL points in D, 
Z_Y_n(x,y)=Z_Y_1(x,y)=+1/2,for ALL points in D, 

(��). 
X_(n+1)=[X_n.Y_n.] 
Z_X_(n+1)(x,y)��X_(n+1)֮Z����, 
��Ҫ��Z_X_n(x,y)��Z_Y_n(x,y)����ʾZ_X_(n+1)(x,y), 

t_i����X axis֮ʱ,i��������, 
t_i is real number and 0С�ڵ���t_iС�ڵ���1 for all i, 
where i is integer and 1С�ڵ���iС�ڵ���m, 
3С�ڵ���m, 
t_1=0,t_m=1, 

t_1С��t_2, 
t_2С��t_3, 
t_3С��t_4, 
...etc... 
t_(m-1)С��t_m, 

(һ). 
��m����������, 
Z_X_(n+1)(x,y)ֻ���������ж��ֺ����е�ĳһ��: 
��һ�ֺ���: 
if 
Z_X_(n+1)(x,y)=Z_X_n((x-t_1)/(t_2-t_1),y), 
where t_1С�ڵ���xС��t_2 and 0С�ڵ���yС��1, 
then 
Z_X_(n+1)(x,y)=Z_X_n((x-t_1)/(t_2-t_1),y), 
where t_1С�ڵ���xС��t_2 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_X_n((x-t_3)/(t_4-t_3),y), 
where t_3С�ڵ���xС��t_4 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_X_n((x-t_5)/(t_6-t_5),y), 
where t_5С�ڵ���xС��t_6 and 0С�ڵ���yС��1; 
...etc... 
and 
Z_X_(n+1)(x,y)=Z_X_n((x-t_(m-2))/(t_(m-1)-t_(m-2)),y), 
where t_(m-2)С�ڵ���xС��t_(m-1) and 0С�ڵ���yС��1; 

and 

Z_X_(n+1)(x,y)=Z_Y_n((x-t_2)/(t_3-t_2),y), 
where t_2С�ڵ���xС��t_3 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_4)/(t_5-t_4),y), 
where t_4С�ڵ���xС��t_5 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_6)/(t_7-t_6),y), 
where t_6С�ڵ���xС��t_7 and 0С�ڵ���yС��1; 
...etc... 
and 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_(m-1))/(t_(m-0)-t_(m-1)),y), 
where t_(m-1)С�ڵ���xС��t_(m-0) and 0С�ڵ���yС��1; 

and 

1*{ [t_2-t_1]+[t_4-t_3]+[t_6-t_5]...etc...+[t_(m-1)-t_(m-2)] }=1/2; 
and 
1*{ [t_3-t_2]+[t_5-t_4]+[t_7-t_6]...etc...+[t_(m-0)-t_(m-1)] }=1/2. 

�ڶ��ֺ���: 
if 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_1)/(t_2-t_1),y), 
where t_1С�ڵ���xС��t_2 and 0С�ڵ���yС��1, 
then 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_1)/(t_2-t_1),y), 
where t_1С�ڵ���xС��t_2 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_3)/(t_4-t_3),y), 
where t_3С�ڵ���xС��t_4 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_5)/(t_6-t_5),y), 
where t_5С�ڵ���xС��t_6 and 0С�ڵ���yС��1; 
...etc... 
and 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_(m-2))/(t_(m-1)-t_(m-2)),y), 
where t_(m-2)С�ڵ���xС��t_(m-1) and 0С�ڵ���yС��1; 

and 

Z_X_(n+1)(x,y)=Z_X_n((x-t_2)/(t_3-t_2),y), 
where t_2С�ڵ���xС��t_3 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_X_n((x-t_4)/(t_5-t_4),y), 
where t_4С�ڵ���xС��t_5 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_X_n((x-t_6)/(t_7-t_6),y), 
where t_6С�ڵ���xС��t_7 and 0С�ڵ���yС��1; 
...etc... 
and 
Z_X_(n+1)(x,y)=Z_X_n((x-t_(m-1))/(t_(m-0)-t_(m-1)),y), 
where t_(m-1)С�ڵ���xС��t_(m-0) and 0С�ڵ���yС��1; 

and 

1*{ [t_2-t_1]+[t_4-t_3]+[t_6-t_5]...etc...+[t_(m-1)-t_(m-2)] }=1/2; 
and 
1*{ [t_3-t_2]+[t_5-t_4]+[t_7-t_6]...etc...+[t_(m-0)-t_(m-1)] }=1/2. 


(��). 
��m��ż������, 
Z_X_(n+1)(x,y)ֻ���������ж��ֺ����е�ĳһ��: 
��һ�ֺ���: 
if 
Z_X_(n+1)(x,y)=Z_X_n((x-t_1)/(t_2-t_1),y), 
where t_1С�ڵ���xС��t_2 and 0С�ڵ���yС��1, 
then 
Z_X_(n+1)(x,y)=Z_X_n((x-t_1)/(t_2-t_1),y), 
where t_1С�ڵ���xС��t_2 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_X_n((x-t_3)/(t_4-t_3),y), 
where t_3С�ڵ���xС��t_4 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_X_n((x-t_5)/(t_6-t_5),y), 
where t_5С�ڵ���xС��t_6 and 0С�ڵ���yС��1; 
...etc... 
and 
Z_X_(n+1)(x,y)=Z_X_n((x-t_(m-1))/(t_(m-0)-t_(m-1)),y), 
where t_(m-1)С�ڵ���xС��t_(m-0) and 0С�ڵ���yС��1; 

and 

Z_X_(n+1)(x,y)=Z_Y_n((x-t_2)/(t_3-t_2),y), 
where t_2С�ڵ���xС��t_3 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_4)/(t_5-t_4),y), 
where t_4С�ڵ���xС��t_5 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_6)/(t_7-t_6),y), 
where t_6С�ڵ���xС��t_7 and 0С�ڵ���yС��1; 
...etc... 
and 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_(m-2))/(t_(m-1)-t_(m-2)),y), 
where t_(m-2)С�ڵ���xС��t_(m-1) and 0С�ڵ���yС��1; 

and 

1*{ [t_2-t_1]+[t_4-t_3]+[t_6-t_5]...etc...+[t_(m-0)-t_(m-1)] }=1/2; 
and 
1*{ [t_3-t_2]+[t_5-t_4]+[t_7-t_6]...etc...+[t_(m-1)-t_(m-2)] }=1/2. 

�ڶ��ֺ���: 
if 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_1)/(t_2-t_1),y), 
where t_1С�ڵ���xС��t_2 and 0С�ڵ���yС��1, 
then 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_1)/(t_2-t_1),y), 
where t_1С�ڵ���xС��t_2 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_3)/(t_4-t_3),y), 
where t_3С�ڵ���xС��t_4 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_5)/(t_6-t_5),y), 
where t_5С�ڵ���xС��t_6 and 0С�ڵ���yС��1; 
...etc... 
and 
Z_X_(n+1)(x,y)=Z_Y_n((x-t_(m-1))/(t_(m-0)-t_(m-1)),y), 
where t_(m-1)С�ڵ���xС��t_(m-0) and 0С�ڵ���yС��1; 

and 

Z_X_(n+1)(x,y)=Z_X_n((x-t_2)/(t_3-t_2),y), 
where t_2С�ڵ���xС��t_3 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_X_n((x-t_4)/(t_5-t_4),y), 
where t_4С�ڵ���xС��t_5 and 0С�ڵ���yС��1; 
and 
Z_X_(n+1)(x,y)=Z_X_n((x-t_6)/(t_7-t_6),y), 
where t_6С�ڵ���xС��t_7 and 0С�ڵ���yС��1; 
...etc... 
and 
Z_X_(n+1)(x,y)=Z_X_n((x-t_(m-2))/(t_(m-1)-t_(m-2)),y), 
where t_(m-2)С�ڵ���xС��t_(m-1) and 0С�ڵ���yС��1; 

and 

1*{ [t_2-t_1]+[t_4-t_3]+[t_6-t_5]...etc...+[t_(m-0)-t_(m-1)] }=1/2; 
and 
1*{ [t_3-t_2]+[t_5-t_4]+[t_7-t_6]...etc...+[t_(m-1)-t_(m-2)] }=1/2. 



(��). 
Y_(n+1)=[Y_n+X_n+] 
Z_Y_(n+1)(x,y)��Y_(n+1)֮Z����, 
��Ҫ��Z_Y_n(x,y)��Z_X_n(x,y)����ʾZ_Y_(n+1)(x,y), 

s_i����Y axis֮��,i��������, 
s_i is real number and 0С�ڵ���s_iС�ڵ���1 for all i, 
where i is integer and 1С�ڵ���iС�ڵ���m, 
3С�ڵ���m, 
s_1=0,s_m=1, 

s_1С��s_2, 
s_2С��s_3, 
s_3С��s_4, 
...etc... 
s_(m-1)С��s_m, 

(һ). 
��m����������, 
Z_Y_(n+1)(x,y)ֻ���������ж��ֺ����е�ĳһ��: 
��һ�ֺ���: 
if 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_1)/(s_2-s_1)), 
where s_1С�ڵ���yС��s_2 and 0С�ڵ���xС��1, 
then 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_1)/(s_2-s_1)), 
where s_1С�ڵ���yС��s_2 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_3)/(s_4-s_3)), 
where s_3С�ڵ���yС��s_4 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_5)/(s_6-s_5)), 
where s_5С�ڵ���yС��s_6 and 0С�ڵ���xС��1; 
...etc... 
and 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_(m-2))/(s_(m-1)-s_(m-2))), 
where s_(m-2)С�ڵ���yС��s_(m-1) and 0С�ڵ���xС��1; 

and 

Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_2)/(s_3-s_2)), 
where s_2С�ڵ���yС��s_3 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_4)/(s_5-s_4)), 
where s_4С�ڵ���yС��s_5 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_6)/(s_7-s_6)), 
where s_6С�ڵ���yС��s_7 and 0С�ڵ���xС��1; 
...etc... 
and 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_(m-1))/(s_(m-0)-s_(m-1))), 
where s_(m-1)С�ڵ���yС��s_(m-0) and 0С�ڵ���xС��1; 

and 

1*{ [s_2-s_1]+[s_4-s_3]+[s_6-s_5]...etc...+[s_(m-1)-s_(m-2)] }=1/2; 
and 
1*{ [s_3-s_2]+[s_5-s_4]+[s_7-s_6]...etc...+[s_(m-0)-s_(m-1)] }=1/2. 

�ڶ��ֺ���: 
if 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_1)/(s_2-s_1)), 
where s_1С�ڵ���yС��s_2 and 0С�ڵ���xС��1, 
then 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_1)/(s_2-s_1)), 
where s_1С�ڵ���yС��s_2 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_3)/(s_4-s_3)), 
where s_3С�ڵ���yС��s_4 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_5)/(s_6-s_5)), 
where s_5С�ڵ���yС��s_6 and 0С�ڵ���xС��1; 
...etc... 
and 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_(m-2))/(s_(m-1)-s_(m-2))), 
where s_(m-2)С�ڵ���yС��s_(m-1) and 0С�ڵ���xС��1; 

and 

Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_2)/(s_3-s_2)), 
where s_2С�ڵ���yС��s_3 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_4)/(s_5-s_4)), 
where s_4С�ڵ���yС��s_5 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_6)/(s_7-s_6)), 
where s_6С�ڵ���yС��s_7 and 0С�ڵ���xС��1; 
...etc... 
and 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_(m-1))/(s_(m-0)-s_(m-1))), 
where s_(m-1)С�ڵ���yС��s_(m-0) and 0С�ڵ���xС��1; 

and 

1*{ [s_2-s_1]+[s_4-s_3]+[s_6-s_5]...etc...+[s_(m-1)-s_(m-2)] }=1/2; 
and 
1*{ [s_3-s_2]+[s_5-s_4]+[s_7-s_6]...etc...+[s_(m-0)-s_(m-1)] }=1/2. 


(��). 
��m��ż������, 
Z_Y_(n+1)(x,y)ֻ���������ж��ֺ����е�ĳһ��: 
��һ�ֺ���: 
if 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_1)/(s_2-s_1)), 
where s_1С�ڵ���yС��s_2 and 0С�ڵ���xС��1, 
then 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_1)/(s_2-s_1)), 
where s_1С�ڵ���yС��s_2 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_3)/(s_4-s_3)), 
where s_3С�ڵ���yС��s_4 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_5)/(s_6-s_5)), 
where s_5С�ڵ���yС��s_6 and 0С�ڵ���xС��1; 
...etc... 
and 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_(m-1))/(s_(m-0)-s_(m-1))), 
where s_(m-1)С�ڵ���yС��s_(m-0) and 0С�ڵ���xС��1; 

and 

Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_2)/(s_3-s_2)), 
where s_2С�ڵ���yС��s_3 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_4)/(s_5-s_4)), 
where s_4С�ڵ���yС��s_5 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_6)/(s_7-s_6)), 
where s_6С�ڵ���yС��s_7 and 0С�ڵ���xС��1; 
...etc... 
and 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_(m-2))/(s_(m-1)-s_(m-2))), 
where s_(m-2)С�ڵ���yС��s_(m-1) and 0С�ڵ���xС��1; 

and 

1*{ [s_2-s_1]+[s_4-s_3]+[s_6-s_5]...etc...+[s_(m-0)-s_(m-1)] }=1/2; 
and 
1*{ [s_3-s_2]+[s_5-s_4]+[s_7-s_6]...etc...+[s_(m-1)-s_(m-2)] }=1/2. 

�ڶ��ֺ���: 
if 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_1)/(s_2-s_1)), 
where s_1С�ڵ���yС��s_2 and 0С�ڵ���xС��1, 
then 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_1)/(s_2-s_1)), 
where s_1С�ڵ���yС��s_2 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_3)/(s_4-s_3)), 
where s_3С�ڵ���yС��s_4 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_5)/(s_6-s_5)), 
where s_5С�ڵ���yС��s_6 and 0С�ڵ���xС��1; 
...etc... 
and 
Z_Y_(n+1)(x,y)=Z_X_n(x,(y-s_(m-1))/(s_(m-0)-s_(m-1))), 
where s_(m-1)С�ڵ���yС��s_(m-0) and 0С�ڵ���xС��1; 

and 

Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_2)/(s_3-s_2)), 
where s_2С�ڵ���yС��s_3 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_4)/(s_5-s_4)), 
where s_4С�ڵ���yС��s_5 and 0С�ڵ���xС��1; 
and 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_6)/(s_7-s_6)), 
where s_6С�ڵ���yС��s_7 and 0С�ڵ���xС��1; 
...etc... 
and 
Z_Y_(n+1)(x,y)=Z_Y_n(x,(y-s_(m-2))/(s_(m-1)-s_(m-2))), 
where s_(m-2)С�ڵ���yС��s_(m-1) and 0С�ڵ���xС��1; 

and 

1*{ [s_2-s_1]+[s_4-s_3]+[s_6-s_5]...etc...+[s_(m-0)-s_(m-1)] }=1/2; 
and 
1*{ [s_3-s_2]+[s_5-s_4]+[s_7-s_6]...etc...+[s_(m-1)-s_(m-2)] }=1/2. 

--
'��������'��'������'��'��?�Űٻ�', 

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��'��������'��X_n,��'��?�Űٻ�'��Y_n, 
��'��������'��Y_n,��'��?�Űٻ�'��X_n, 

��?�������׶ȷ����������ʵ�ṹ(��������)������Ԫ2002��, 
��'��?�Űٻ�'��'��������'����Ԫ2002��ͬʱ���ַ�����������(�������ʵ�ṹ), 
���巢����05-20-2002, 

--

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