Seifert-Weber

# Seifert-Weber Space for the HP-41

Overview

-Hyperbolic spaces are usually infinite... if they are not multiconnected !

-The Seifert-Weber dodecahedral space is an example of a 3-Dimension hyperbolic space with a constant curvature R and a finite volume V.
-It is obtained by gluing the pairs of opposite sides of a regular hyperbolic dodecahedron, with a twist of 108°.

-If R = 1, this volume may be computed by the formula:

V = 30 [ 2 L(36°+d) - 2 L(36°-d) - L(30°+d) + L(30°-d) + 2 L(90°-d) ]   where  d = Arc Tan [ sqrt(-14+10.sqrt(5)) / (3+sqrt(5)) ]

and  L = Lobachevsky Function.

Program Listing

Data Registers:  R00 to R02: temp  R01 = Volume  at the end
Flags: /
Subroutine:  "LOB"  Lobachevsky function ( cf "Lobachevsky Function for the HP-41" )

 01  LBL "SWV"  02  DEG  03  304  04  500  05  SQRT           06  14  07  +  08  /  09  SQRT  10  5 11  SQRT  12  3  13  +  14  /  15  ATAN  16  STO 02         17  36  18  +  19  XEQ "LOB"  20  STO 01 21  36  22  RCL 02         23  -  24  XEQ "LOB"  25  ST- 01  26  90  27  RCL 02   28  -  29  XEQ "LOB"  30  RCL 01 31  +  32  ST+ X  33  STO 01   34  30  35  RCL 02         36  +  37  XEQ "LOB"  38  ST- 01  39  30  40  RCL 02 41  -  42  XEQ "LOB"  43  RCL 01         44  +  45  30  46  *  47  STO 01  48  END

( 91 bytes / SIZE 003 )

 STACK INPUT OUTPUT X / V

With  V = 11.19906475   ( the last digit should be a 4 )

Notes:

-If R # 1, simply multiply the result by R^3.

-Lines 03 to 15 may be replaced by a single line  28.90854705
-And the whole program may be reduced to

 01  LBL "SWV"  02  11.19906474  03  END

-Or, if you place R in X-register:

 01  LBL "SWV"  02  3  03  Y^X  04  11.19906474  05  *  06  END

-By comparison, the volume of a 3D-hypersphere is 2(PI)^2 = 19.73920880
-So, the Seifert-Weber space is smaller than the hypersphere.

Reference:

[1]  John G. Ratcliffe - "Foundations of Hyperbolic Manifolds" -  which may be dowloaded freely  here