***Finding Seked Angles of Ancient Egypt, 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 finding seked angles of ancient egypt for the etcl console. After you read the Rhind papyrus, an ancient Egyptian asks you for a causeway of 12 palms to the pyramid. What is the angle of the causeway 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 %|quantity|units|ratio |angle| units|method|% &|8.0 |palms | 7/8 | 41.1|degrees|ti30 hand calculation|& &|9.0 |palms| 7/9 | 37.87|degrees|ti30 hand calculation|& &|10.0 |palms| 7/10 | 34.9|degrees|ti30 hand calculation|& &|11.0|palms | 7/11 | 32.47|degrees|ti30 hand calculation|& &|12 | palms| 7/12 | 30.2|degrees|ti30 hand calculation|& &|13 | palms| 7/13 | 28.3|degrees|ti30 hand calculation|& &|14 | palms| 7/14 | 26.56|degrees|ti30 hand calculation|& &|15 | palms| 7/15 | 25.0|degrees|ti30 hand calculation|& &|20:3 | palms:fingers| 7/(20+3/4) | 18.6|degrees|goggle paste calculation|& &|25:3 | palms:fingers| 7/(25+3/4) | 15.2|degrees|goggle paste calculation|& &|40:3 | palms:fingers| 7/(40+3/4) | 9.74|degrees|goggle paste calculation|& &|70:3 | palms:fingers| 7/(70+3/4) | 5.65|degrees|goggle 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) = 30.2564372, 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))) = 28.7676493. ---- 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 or implied slope is of interest here. One can paste a conditional answer in google as (730. * 55. * 30.) / (.725 * .362 * .244) = 18.8E6 bricks. 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. 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 had a slope of 1:10 or 0.1. This equates to an angle of 5.7 degrees, (180. / pi) * atan(1 / 10) = 5.71059314. From our calculations, a seked of 70:3 palms:fingers would have an angle of 5.65 degrees with the horizon. ---- ***Screenshots Section*** ---- Due size of following jpgs, leaving pictures as point and click. * http://commons.wikimedia.org/wiki/File:Rhind_Mathematical_Papyrus.jpg * http://en.wikipedia.org/ wiki/ File:Geometrical_diagram_showing_Egyptian_seked_system.jpg * http://en.wikipedia.org/wiki/File:Geometrical_diagram_showing_Egyptian_seked_system.jpg * http://commons.wikimedia.org/wiki/File:Egyptian_A%27h-mos%C3%A8_or_Rhind_Papyrus_(1065x1330).png ---- '''Comments Section''' Please place any comments here, Thanks. ---- ***References:*** * http://www.kch42.dial.pipex.com/sekes.htm * http://www.math.washington.edu/~greenber/pyressay.html * http://www.google.com/intl/en/help/features.html#calculator * http://en.wikipedia.org/wiki/Seked * http://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus * http://en.wikipedia.org/wiki/Ancient_Egyptian_units_of_measurement * [Angles on a compass] ---- ****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 # output plotted in jpegs # code from TCL WIKI, eTCL console script # 8jun2011, [gold] console show proc errorx {aa bb} {expr { $aa > $bb ? (($aa*1.)/$bb -1.)*100. : (($bb*1.)/$aa -1.)*100.}} proc radianstodegconst {} {return [ expr {180./[pi]} ]} proc xradianstodegconst {} {return [ expr {[pi]/180.} ]} proc degz {} {return [ expr {180./[pi]} ]} proc degx {aa} {return [ expr { [degz]*atan($aa) } ]} 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 $angle1 " incr counter wm title . "estimating seked (cotan ratio)" } seked 7 12 ====== *** Code scraps*** ====== puts "$counter height $aa base $bb angle $angle1 " ====== <> Testing | Toys | Physics | Games |Statistics| Example | Mathematics