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