File:Domino effect visualizing exclusion of junk term by induction axiom.jpg

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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.

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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I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
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Date/TimeThumbnailDimensionsUserComment
current16:55, 30 October 2013Thumbnail for version as of 16:55, 30 October 20133,200 × 2,400 (648 KB)Jochen Burghardt (talk | contribs)User created page with UploadWizard

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