Finding Seked Angles of Ancient Egypt, Console Example

Finding Seked Angles of Ancient Egypt, Console Example


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Introduction

gold Here is an eTCL script on finding seked angles of ancient egypt for the etcl console. After you read the Rhind papyrus, an ancient Egyptian asks you for a roof of 12 palms for his house. What is the angle of the roof in degrees?


In planning any software, there is a need to develop testcases. With back of envelope calculations, we can develop a number of peg points to check output of program.


Testcases


seked numbers and ratios

quantityunitsratio angle unitsmethod
8.0 palms 7/8 41.1degreesti30 hand calculation
9.0 palms 7/9 37.87degreesti30 hand calculation
10.0 palms 7/10 34.9degreesti30 hand calculation
11.0palms 7/11 32.47degreesti30 hand calculation
12 palms 7/12 30.2degreesti30 hand calculation
13 palms 7/13 28.3degreesti30 hand calculation
14 palms 7/14 26.56degreesti30 hand calculation
15 palms 7/15 25.0degreesti30 hand calculation
20:3 palms:fingers 7/(20+3/4) 18.6degreesgoggle paste calculation
25:3 palms:fingers 7/(25+3/4) 15.2degreesgoggle paste calculation
40:3 palms:fingers 7/(40+3/4) 9.74degreesgoggle paste calculation
70:3 palms:fingers 7/(70+3/4) 5.65degreesgoggle paste calculation

A seked is what moderns might call the cotangent ratio of a right triangle. The vertical side is a one cubit rod of 7 palms height and the base is some measure of palms and fingers. In this problem set, the cubit rod is fixed at 7 palms high and each palm measure of the base can be divided into 4 fingers. For example, a seked measurement might be given as 12 palms and 3 fingers. The formula for a seked ratio of 7/12 is (180 / pi) * atan(7 / 12) = 30.2 degrees, which can be pasted into google search engine and solve. The formula for a seked ratio of 7 palms /12 palms 3 fingers is (180. / pi) * atan(7 / (12 + (3 / 4))) = 28.7 degrees.


In the Ahmes papyrus of 1650 BCE, a pyramid with a height of 250 cubits and base of 360 cubits was found to have a seked of 5+1/25 or 5.04 palms. One can paste (7 * .5 * 360) / 250 = 5.04 in decimal units.


For the testcases, one would like to get realistic testcases from the ancient sources. The gist of a scribal copy exercise (1200 BCE.) formulates a ramp as 730 cubits,55 cubits wide,60 cubits high, and 30 cubits high in the middle and asks how many bricks are needed. The implied seked ratio, angle, and slope is of interest here. One can postulate an Egyptian mud brick as 0.725 cubit long, 0.362 cubit wide, and 0.244 cubit height. One can paste a conditional answer in google as (730. * 55. * 30.) / (.725 * .362 * .244) = 18.8E6 bricks. For the angle, one can paste (180. / pi) * arctan(60. / 730.) = 4.69 degrees. In terms of the TCL console program the units cancel out, so the command seked can be entered as seked 30 730 (cubits). The TCL console program calculates the seked ratio as 7:70 and the angle as 4.7 degrees. Dividing the ratio 7:70 by 7, gives a slope of 1:10.

   console session
  1 height 60  base  730  angle  4.69868051729944 complement 85.30131948270056  
  bricks 18809203.314292498
  rev seked  7: 70.0 
  sek 70.0

In modern terms, the causeway to the great pyramid might have had a slope of 1:10 or 0.1. This equates to an angle of 5.7 degrees, (180. / pi) * atan(1 / 10) = 5.7 degrees. From our calculations, a seked of 70:3 palms:fingers would have an angle of 5.65 degrees with the horizon. From the proportions at Giza and other conjectures, the causeway is postulated to be 430 meters long, 43 meters high, and 10 meters wide. For the angle, one can paste (180. / pi) * arctan(43. / 430.) = 5.7 degrees. Using the above mud brick as a concept, one could paste (.5 * (430. * 43. * 10.)) / (.38 * .19 * .128) = 10E6 bricks. For stone blocks, there might be (.5 * (430. * 43. * 10.)) / (.8 * .8 * 1.6) = 9E4 blocks.


For an estimate of the number of bricks or stone blocks, one can use pseudocode and some one line procs from Oneliner's Pie in the Sky.

 pseudocode: number of blocks = volume of structure over volume of block
 proc rectangularprismvolumex {aa bb cc }  { return [ expr { $aa*$bb*$cc  }] };# from [Oneliner's Pie in the Sky]
 pseudocode: proc bricks { aa bb cc } { volume of structure over volume of block;return} 
 pseudocode: The ramp volume is half a rectangular prism volume.
 pseudocode: proc bricks { aa bb cc } { .5* [rectangularprismvolumex $aa $bb $cc] over rectangularprismvolume .8   .8   1.6 }
 proc bricks { aa bb cc } { 
  set struct [ rectangularprismvolumex $aa $bb $cc ]
  set brick  [ rectangularprismvolumex .8   .8   1.6x ]
  set results [ expr { (0.5*$struct) / $brick } ]
  return $results }


Due size of following jpgs, leaving pictures as point and click.


References:


Appendix TCL programs and scripts

* Pretty Print Version

        # Pretty print version from autoindent
        # and ased editor
        # written on Windows XP on eTCL
        # program :estimating seked (cotan ratio) and angles
        # from egyptian pyramids
        # code from TCL WIKI, eTCL console script
        # 8jun2011, [gold]
        console show 
        proc radianstodegconst {} {return [ expr {180./[pi]}  ]}
        proc pi {} {expr acos(-1)}
        set counter 1
        set past 0
        proc seked { aa bb  } {
            global counter past
            set angle1 [ expr { ($aa*1.)/($bb*1.) } ]
            set angle1 [ expr { atan($angle1) } ]
            set angle1 [ expr { $angle1* [radianstodegconst]} ]
            puts "$counter height $aa  base  $bb  angle  $angle1   "
            incr counter
            wm title . "estimating seked (cotan ratio)"
        }
        seked 7 12

Code scraps

 console show
    proc rectangularprismvolumex {aa bb cc }  { return [ expr { $aa*$bb*$cc  }] }
    proc bricks { aa bb cc } {
        set struct [ rectangularprismvolumex $aa $bb $cc ]
        set brick  [ rectangularprismvolumex .8   .8   1.6 ]
        set results [ expr { (.5*$struct) / $brick } ]
        return $results }
    puts " bricks [ bricks 430.   43.   10.] "     

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