Updated 2017-03-09 23:18:53 by gold

## Introduction edit

gold Here is some eTCL starter code for calculating the volume of a truncated rectangular pyramid. Most of the testcases involve replicas or models, using assumptions and rules of thumb.

Neugebauer, Robson, and Friberg, J. published cuneiform math problems which computed the volume of a truncated triangular prism, rectangular prism, and other Sumerian geometric figures. Since there is some uncertancy in the Sumerian geometric volumes, the calculator is starting and adapting fron the truncated rectangular pyramid in the modern formula. I have modified a slot calculator in eTCL to handle calculations for the rectangular pyramid volume. At this point, its easier to use modern geometric equations in the eTCl calculator and compute the volume accurately. Then try to understand how the Sumerians developed their geometric figures and coefficients. Although the geometric coefficients were passed through successive cultures as late as 400 BCE, the main compilation of the geometric coefficients was probably during the King Naram-Sin reforms of the Akkadian empire in 2150 BCE.

### Pseudocode and Equations

```    # following statements can be pasted into eTCL console
set testcase_1  [ eval expr  ( 1+1 +1+1)/4 ]  # 1 square meters
set testcase_2  [ eval expr  ( 2+2 +4+4)/4 ]  # 3 square meters
set testcase_3  [ eval expr  ( 3+3 +5+5)/4 ]  # 4 square meters
set testcase_1_perimeter   [ eval expr  ( 1+1 +1+1) ]  # 4 square meters
set testcase_2_perimeter   [ eval expr  ( 2+2 +4+4) ]  # 12 square meters
set testcase_3_perimeter   [ eval expr ( 3+3 +5+5)  ]  # 16 square meters```

### Testcases Section

In planning any software, it is advisable to gather a number of testcases to check the results of the program. The math for the testcases can be checked by pasting statements in the TCL console. Aside from the TCL calculator display, when one presses the report button on the calculator, one will have console show access to the capacity functions (subroutines).

#### Testcase 1

table 1printed in tcl wiki format
quantity value comment, if any
testcase number:1
1.0 :length L1 meters meters
1.0 :width W1 meters
1.0 :top T1 meters
1.0 :width W2 meters
1.5 :optional, conversion units coefficient
1.5 :converted volume cubic units
1.0 :volume cubic meters

#### Testcase 2

table 2printed in tcl wiki format
quantity value comment, if any
testcase number:2
2.0 :length L1 meters meters
2.0 :width W1 meters
4.0 :top T1 meters
4.0 :width W2 meters
1.5 :optional, conversion units coefficient
24.0 :converted volume cubic units
16.0 :volume cubic meters

#### Testcase 3

table 3printed in tcl wiki format
quantity value comment, if any
testcase number:3
10.0 :length L1 meters meters
6.0 :width W1 meters
4.0 :top T1 meters
4.0 :width W2 meters
1.5 :optional, conversion units coefficient
471.99999999999994 :converted volume cubic units
314.66666666666663 :volume cubic meters

### References:

• Eleanor Robson, Mesopotamian Mathematics, 2100-1600 BC (Oxford, 1999)
• Robson, Eleanor, Mesopotamian Mathematics, 2100-1600BCE,Oxford 1999
• Horowitz, Wayne, Late Babylonian Tablet CBS1766, Hebrew University
• Steele, J.M. Celestial Measurement in Bablylonian Astronomy,Annals of Science,2007
• Mathematical Cuneiform Texts, Neugebauer and A. Sachs, American Oriental Society, 1945
• Friberg 1987-90:555,Firberg on tablet BM15285
• Eclipse Prediction and the Length of the Saros in
• Babylonian Astronomy LIS BRACK-BERNSEN∗AND JOHN M. STEELE
• Celestial Measurement in Babylonian Astronomy, J. M. STEELE, University of Durham
• Amazing Traces of a Babylonian Origin in Greek Mathematics, Jöran Friberg and Joachim Marzahn
• The area and the side i added: some old Babylonian geometry, duncan j. Melville
• Sumerian Circular Segment Coefficients and Calculator Demo Example
• Sumerian Coefficients in the Pottery Factory and Calculator Demo Example
• Sumerian Pottery Vessel & Clay Mass and eTCL Slot Calculator Demo Example , numerical analysis
• Mathematics hidden behind the two coefficients of Babylonian geometry, kazuo muroi
• Especially oven/kiln problem in YBC7997, area of ring annulus complements bullseye figure.
• Sumerian Barge & Cargo Calculator and eTCL Slot Calculator Demo Example, numerical analysis
• Sumerian Coefficients at the Weavers Factory and eTCL Slot Calculator Demo Example
• Sumerian Construction Rates and eTCL Slot Calculator Demo Example
• Sumerian Workcrew & Payroll and eTCL Slot Calculator Demo Example, numerical analysis
• The design of Babylonian waterclocks : Astronomical and experimental evidence,Steele

## Appendix Code edit

### appendix TCL programs and scripts

```        # pretty print from autoindent and ased editor
# Sumerian Pyramid Formula calculator
# written on Windows XP on eTCL
# working under TCL version 8.5.6 and eTCL 1.0.1
# gold on TCL WIKI, 15may2016
package require Tk
namespace path {::tcl::mathop ::tcl::mathfunc}
frame .frame -relief flat -bg aquamarine4
pack .frame -side top -fill y -anchor center
set names {{} {length L1 meters :} }
lappend names {width W1 meters :}
lappend names {height H1 meters: }
lappend names {optional, top T1 meters :}
lappend names {optional, conversion units coefficient:}
lappend names {answers: volume cubic meters: }
lappend names {converted volume cubic units: }
lappend names {volume cubic meters :}
foreach i {1 2 3 4 5 6 7 8} {
label .frame.label\$i -text [lindex \$names \$i] -anchor e
entry .frame.entry\$i -width 35 -textvariable side\$i
set msg "Calculator for Sumerian Pyramid
from TCL WIKI,
written on eTCL "
tk_messageBox -title "About" -message \$msg }
proc pi {} {expr acos(-1)}
proc calculate {     } {
global side1 side2 side3 side4 side5
global side6 side7 side8
global testcase_number
global average_length average_width total_surface_area
incr testcase_number
set side1 [* \$side1 1. ]
set side2 [* \$side2 1. ]
set side3 [* \$side3 1. ]
set side4 [* \$side4 1. ]
set side5 [* \$side5 1. ]
set side6 [* \$side6 1. ]
set side7 [* \$side7 1. ]
set side8 [* \$side8 1. ]
set L1 \$side1
set T1 \$side2
set H1 \$side3
set conv_factor \$side5
set sq_pyramid_vol [* \$H1   [+   [* \$L1 \$L1 ]  [* \$T1 \$T1 ]  [* \$L1 \$L1 ] ] [/ 1. 3.] ]
set conv_pyramid_vol [* \$sq_pyramid_vol \$conv_factor ]
set side6 \$sq_pyramid_vol
set side7 \$conv_pyramid_vol
set side8 \$sq_pyramid_vol
}
proc fillup {aa bb cc dd ee ff gg hh} {
.frame.entry1 insert 0 "\$aa"
.frame.entry2 insert 0 "\$bb"
.frame.entry3 insert 0 "\$cc"
.frame.entry4 insert 0 "\$dd"
.frame.entry5 insert 0 "\$ee"
.frame.entry6 insert 0 "\$ff"
.frame.entry7 insert 0 "\$gg"
.frame.entry8 insert 0 "\$hh"
}
proc clearx {} {
foreach i {1 2 3 4 5 6 7 8 } {
.frame.entry\$i delete 0 end } }
proc reportx {} {
global side1 side2 side3 side4 side5
global side6 side7 side8
global testcase_number
global average_length average_width total_surface_area
console show;
puts "%|table \$testcase_number|printed in| tcl wiki format|% "
puts "&| quantity| value| comment, if any|& "
puts "&| testcase number:|\$testcase_number | |&"
puts "&| \$side1 :|length L1 meters meters |   |&"
puts "&| \$side2 :|width W1 meters | |& "
puts "&| \$side3 :|top T1 meters| |& "
puts "&| \$side4 :|width W2 meters| |&"
puts "&| \$side5 :|optional, conversion units coefficient | |&"
puts "&| \$side6 :|answers: volume cubic meters |  |&"
puts "&| \$side7 :|converted volume cubic units|  |&"
puts "&| \$side8 :|volume cubic meters|  |&"
}
frame .buttons -bg aquamarine4
::ttk::button .calculator -text "Solve" -command { calculate   }
::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 1.0 1.0 1.0 1.0  1.50 1.0 1.0 1.0}
::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 2. 2. 4.0 4.0   1.50  2.0 3.0 3.0 }
::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 10. 6. 4.0 4.0   1.50 2.0  4.0 4.0 }
::ttk::button .clearallx -text clear -command {clearx }
::ttk::button .cons -text report -command { reportx }
::ttk::button .exit -text exit -command {exit}
pack  .clearallx .cons .about .exit .test4 .test3 .test2   -side bottom -in .buttons
grid .frame .buttons -sticky ns -pady {0 10}
. configure -background aquamarine4 -highlightcolor brown -relief raised -border 30
wm title . "Sumerian Pyramid Formula Calculator"```

### Pushbutton Operation

For the push buttons, the recommended procedure is push testcase and fill frame, change first three entries etc, push solve, and then push report. Report allows copy and paste from console.

For testcases in a computer session, the eTCL calculator increments a new testcase number internally, eg. TC(1), TC(2) , TC(3) , TC(N). The testcase number is internal to the calculator and will not be printed until the report button is pushed for the current result numbers. The current result numbers will be cleared on the next solve button. The command { calculate; reportx } or { calculate ; reportx; clearx } can be added or changed to report automatically. Another wrinkle would be to print out the current text, delimiters, and numbers in a TCL wiki style table as
```  puts " %| testcase \$testcase_number | value| units |comment |%"
puts " &| volume| \$volume| cubic meters |based on length \$side1 and width \$side2   |&"  ```