File:Domino effect visualizing exclusion of junk term by induction axiom.jpg
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Summary
| Description |
English: Shows an infinite chain of (light wood) domino pieces and a circle of (dark wood) pieces. If the first light piece is overthrown, each light piece will eventually fall, while no dark piece will be affected. The shown configuration illustrates a model of Peano's axioms for natural numbers, except for the induction axiom. The latter requires all pieces to fall if the first one is overthrown.
A corresponding formal model uses the set ℕ ∪ { di : 0 ≤ i < 90 }, where each di is a fresh constant. One defines 0 and the successor function s on ℕ as usual, and s(di) = di+1 for 0 ≤ i < 89, and s(d89) = d0.
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| Date | |
| Source | Own work, inspired by File:Dominoeffect.png |
| Author | Jochen Burghardt |
| Povray source code |
|---|
// for rendering, use commands:
// povray -Q9 +A +UL +UV +W3200 +H2400 -Idomino.pov
// pngtopnm domino.png | pnmtojpeg >domino.jpg
#include "colors.inc"
#include "stones.inc"
#include "woods.inc"
#declare camX = <-30,0,0>; // observer's left-right offset
camera { location camX+<0,-50,-18> look_at camX+<0,0,-10> }
light_source { camX+<150,-100,-350> color White }
light_source { camX+<1500,-1500,-50000> color rgb<0.2,0.2,0.2> }
// sky
background { color rgb <.7, .7, 1> }
// ground
plane { <0,0,1>, 10.0 texture { T_Stone19 scale 20 } }
// a single domino stone
#declare piece = box { <0,0,2>, <4,0.5,10> }
// "infinite" chain, reachable by induction
#declare radB = <3600,0,0>; // radius of big circle segment
#declare i=0;
#while(i<1100)
object { piece
translate radB
rotate <0,0,i*0.07>
translate -radB
texture { T_Wood10 scale 5 }
}
#declare i=i+1;
#end
// junk terms, not reachable by induction
#declare radW = <40,0,0>; // radius of small circle
#declare i=0;
#while(i<90)
object { piece
translate radW
rotate <0,0,i*4>
translate -radW
translate <-15,70,0>
texture { T_Wood2 scale 5 }
}
#declare i=i+1;
#end
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| Annotations | This image is annotated: View the annotations at Commons |
File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 16:55, 30 October 2013 | | 3,200 × 2,400 (648 KB) | Jochen Burghardt | User created page with UploadWizard |
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