Version 105 of Chinese Sun Stick Accuracy for Console Example

Chinese Sun Stick Accuracy for Console Example

This page is under development. Comments are welcome, but please load any comments in the comments section at the middle of the page. Thanks, gold

gold Here is an eTCL script on Chinese Sun Stick Accuracy. I have modified a console program in eTCL to handle calculations for Sun Stick Accuracy. The sun stick is called a gnomon in the west and a gui-baio in China. There are a number of issues that have risen with the accuracy of the sun stick. In the west, the gnomon was primarily used in the sundial, hence the issue is the accuracy of keeping time with the sundial. In China, the sun stick was used originaly in creating the Chinese calender, in marking and orienting building and towns, and then in map making for the empire. Sun sticks are still being made in classrooms aroung the world to estimate latitude and the size of the earth.

In planning any software, it is advisable to gather a number of testcases to check the results of the program. For the sun stick, these cases represent a diverse lot spanning the globe and several thousand years of history. Shadow diffrusion or shadow blur is a problem in the larger gnonoms. The large gnonom at Jaipur has a shadow blur of 100 mm. The solar disk has a apparent size of 0.5 degree and edge instruments will have an inherent error due to the extended light source. However, the large gnonom at Jaipur, the Duofeng-Xian instrument, and other Yuan period gui-biao instruments had special pointing needles, metal reticles, pinhole cameras, mathematical averaging of the results, and other gismos to alleviate the shadow diffusion. Also, the perpendicular geometry of the gnonom has to be correct and the shadow table level.

Chinese Han text.

Arithmetic Classic of the Gnomon and the Circular Paths of Heaven reported triangle as follows.

    60000 li, 24948 km, 15502.2miles
    80000 li, 33264 km, reputed altitude of sun,
    100000 li ,41580km, reputed distance from measuring  stick to  sun
    Chinese li was 0.4158 km (0.25837 miles) in Han times.
    Modern li is 0.5 km.
    2* babylonian 12709 =25418

On gnomons, height translates into accuracy of latitude measurement or year measurement. Chinese sun/pole star angles of 12 meters +- 2millimeters. For example inferred tangent accuracy from gnomon might be height of gnomon over length of ( shadow +- 2 millimeters). Within reason, the higher the temple or pyramid gnomon and longer the step path, the more accurate the gnomon and measurement of year. Thats why the Mayans et al built temples with "two towers" on top.

In China, Guo Shoujing of Yuan dynasty a remarkably large gnonom in 1279CE. Gnomon are called gui-biao in china. After checking the figures of the large gnomon, an error analysis program for estimating the error or sun angle was written as a console script in eTCL The biao or gnomon head was 12.62 meters, The gui or horizontal measurement table was 128 chi or 31.196 meters.

From the dimensions of Guo's large gnomon, the worst case accuracy was

    pseudocode:((12.62m +-2mm) / (31.196  +-2mm)
    pseudocode:(12.62m -2mm) / (31.196  +2mm)>
                12.62/31198.>  0.312481569
    pseudocode:(12.62m +2mm) / (31.196  -2mm) >
                12620/31194.> 0.4046550

    pseudocode: 22.0309 deg versus  22.0220deg,
    diff  0.00886  deg
    circumference=distance between points*360/angular separation

As a check from non-TCL sources, Guo calculated the tropical year to be 365.2425 days and the reported error of winter solstice was 0.01 day. (Meaning the stick calculations here do not take into account multiple measurements and averaging techniques.)

Other gnomons were built during the Yuan era in China. The heads in China were 8 chi or 1.96 meters. The gui or horizontal tray were about 6.72 meters. Presumably the measurement error was still about 2 mm.

The chief formula that came from the Han Chinese was "cun qian li" , one cun per sun stick for 1000 li. This formula relates distance on earth to change in the length of the sun stick's shadow at noon. Expressed in modern terms, the modern constant would be 40068 km /360 degrees or 111.3 km per degree, based on the earth's circumference of 40,068 km. Unfortunately, the stick measures seem to have changed since Han times. On the Han maps, there are distances given between cities, source not given. For the modern li of 0.5 kilometers, the modern constant would be 222.6 li per degree. Based on the sunstick calculator, 2-5 percent accuracy would about what could be expected from a Han sunstick. Taking half the solar disk as the upper bound on accuracy(eg. 0.25 degree), the upper bound on the unaided sunstick would be about [(45+.25)/(45-.25)]-1.]*100. or 1 percent.

     ***Screenshots Section***

Due to size of gifs, leaving them as point and click

 [http://www.arthursclipart.org/science/science/sundial.gif]

   [http://www.arthursclipart.org/fromthepast/past/aztec%20temple.gif]

Jantar Mantar,Jaipur,India

Karnak, Egypt

Comments Section

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References:

Appendix Code

appendix TCL programs and scripts

*************

FIRST VERSION

 #start of deck
 #start of deck
 #start of deck  
 #start of deck
 #start of deck

 # estimating errors for sun angles
 # code from TCL WIKI, eTCL console script    
 # 6Aug2010, [gold]

 console show
 proc errorx  {aa bb} {
   expr { $aa > $bb ?   (($aa*1.)/$bb -1.)*100. : (($bb*1.)/$aa -1.)*100.}
 } 

 proc degtoradiansconst {} {
   return [ expr {180./[pi]}  ]
 }

 proc degz {} {
   return [ expr {180./[pi]}  ]
 }

 proc degx {aa} {
   return [ expr { [degz]*atan($aa) }  ]
 }

 proc pi {} {
   expr acos(-1)
 }

 proc gm { aa bb cc } {
   set side1 [ expr { ($aa+$cc*1.)/( $bb-$cc*1.)} ] 
   set side2 [ expr { ($aa-$cc*1.)/( $bb+$cc*1.)} ] 
   set side3 [ errorx $side1 $side2 ] 
   set angle1 [ degx  $side1  ]
   set angle2 [ degx  $side2  ]
   set anglediff [ expr { abs($angle1 - $angle2)  } ]       
   puts "$aa height in mm  "
   puts "$bb width in mm  "
   puts "$cc length error in mm "
   puts "$side1 ratio 1 "
   puts "$side2 ratio 2 "
   puts "$angle1 tan 1 in deg"
   puts "$angle2 tan 2 in deg "
   puts "$anglediff  diff  tans  in deg "
   puts "$side3 percent error "
 }

 gm 1960 6270 2
 gm 9746.8 31196 2

 1960 height in mm  
 6270 width in mm  
 2 length error in mm 
 0.31301850670070197 ratio 1 
 0.3121811224489796 ratio 2 
 17.381085802796047 tan 1 in deg
 17.337378262554374 tan 2 in deg 
 0.043707540241673115  diff  tans  in deg 1960 height in mm  
 6270 width in mm  
 2 length error in mm 
 0.31301850670070197 ratio 1 
 0.3121811224489796 ratio 2 
 17.381085802796047 tan 1 in deg
 17.337378262554374 tan 2 in deg 
 0.043707540241673115  diff  tans  in deg 
 0.26823667144038055 percent error 

 gm 12620 31194 2

 12620 height in mm  
 31194 width in mm  
 2 length error in mm 
 0.40465503975378303 ratio 1 
 0.4044749326836774 ratio 2 
 22.030965750860332 tan 1 in deg
 22.0220978158892 tan 2 in deg 
 0.00886793497113203  diff  tans  in deg 
 0.04452861118255935 percent error 
 1% 

 #end of deck
 #end of deck
 #end of deck      
 #end of deck
 #end of deck
 ----
Results of Gui-biao,Beijing,china 
gm 1960 6270 2#entry
1960 height in mm  
6270 width in mm  
2 length error in mm 
0.31301850670070197 ratio 1 
0.3121811224489796 ratio 2 
17.381085802796047 tan 1 in deg
17.337378262554374 tan 2 in deg 
0.043707540241673115  diff  tans  in deg 
0.26823667144038055 percent error 
----
Results of Dengfeng-Xian,Henan,china 
gm 9746.8 31196 2 #entry
9746.8 height in mm  
31196 width in mm  
2 length error in mm 
0.3125216387766878 ratio 1 
0.3123533559843579 ratio 2 
17.355154136448714 tan 1 in deg
17.34636974983224 tan 2 in deg 
0.00878438661647607  diff  tans  in deg 
0.05387577533768617 percent error 
----
Results of Temple of Sun, Teotihuacan site, Mexico
gm 71200. 894000. 200. #entry
71200. height in mm  
894000. width in mm  
200. length error in mm 
0.07988364287312598 ratio 1 
0.07940058152538582 ratio 2 
4.5672968223805075 tan 1 in deg
4.539793893821857 tan 2 in deg 
0.027502928558650552  diff  tans  in deg 
0.6083851509144367 percent error 
----
Results of Tikal Temple, Tikal site, Mexico
gm 47000. 250000. 200. #entry
47000. height in mm  
250000. width in mm  
200. length error in mm 
0.188951160928743 ratio 1 
0.18705035971223022 ratio 2 
10.699955707141847 tan 1 in deg
10.594765712337136 tan 2 in deg 
0.1051899948047108  diff  tans  in deg 
1.0161975734433781 percent error 
----