OEIS/FASS curves: Difference between revisions

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[[File:Draht-Kn.jpg|250px|right|Steel wire]]
===Beyond [http://oeis.org/search?q=A220952 OEIS sequence A220952]===
===Beyond [http://oeis.org/search?q=A220952 OEIS sequence A220952]===
[[File:Fischertechnik-Kn.jpg|250px|right|Fischertechnik 15 mm]]
Sequence A220952 is best discussed with numbers to base 5. The first 125 values then are:
Sequence A220952 is best discussed with numbers to base 5. The first 125 values then are:
  0,
  0,
  1,2,3,4,
  1,2,3,4,
  14,24,34,33,32,31,21,22,23,13,12,11,10,20,30,40,41,42,43,44,
  14,24,34,33,32,31,21,22,23,13,12,11,10,20,30,40,41,42,43,44,
  144,244,344,343,342,341,241,242,243,143,142,141,140,240,340,330,320,310,210,220,
  144,244,344,343,342,341,241,242,243,143,142,141,140,240,340,
230,130,120,110,111,112,113,123,122,121,131,132,133,233,232,231,221,222,223,213,
330,320,310,210,220,230,130,120,110,111,112,113,123,122,121,
212,211,311,312,313,323,322,321,331,332,333,334,324,314,214,224,234,134,124,114,
131,132,133,233,232,231,221,222,223,213,212,211,311,312,313,
  104,204,304,303,302,301,201,202,203,103,102,101,100,200,300,400,401,402,403,404,
323,322,321,331,332,333,334,324,314,214,224,234,134,124,114,
414,424,434,433,432,431,421,422,423,413,412,411,410,420,430,440,441,442,443,444
  104,204,304,303,302,301,201,202,203,103,102,101,100,200,300,
400,401,402,403,404,414,424,434,433,432,431,421,422,423,413,
412,411,410,420,430,440,441,442,443,444
In each generation the number of digits is increased by one. The first 25 values form the adjacency condition matrix as defined by Knuth:
In each generation the number of digits is increased by one. The first 25 values form the adjacency condition matrix as defined by Knuth:
  04==14==24==34  44
  04==14==24==34  44
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  ||  ||          ||
  ||  ||          ||
  00  10==20==30==40
  00  10==20==30==40
[[File:Balsaholz-8mm-Kn.jpg|250px|right|Balsa lumber 8 mm]]
Here, the last digit is the y coordinate, and the digit before is the x coordinate. In generation 3, the left-most digit can be interpreted as the z coordinate. Since this is Knuth's original adjacency matrix, and since it shows a letter "n", I named it "Kn".
Here, the last digit is the y coordinate, and the digit before is the x coordinate. In generation 3, the left-most digit can be interpreted as the z coordinate. Since this is Knuth's original adjacency matrix, and since it shows a letter "n", I named it "Kn".
===The curve is '''[https://en.wikipedia.org/wiki/Space-filling_curve space-filling]''' in every dimension===  
===The curve is '''[https://en.wikipedia.org/wiki/Space-filling_curve space-filling]''' in every dimension===  

Revision as of 19:48, 10 September 2017

Steel wire
Steel wire

Beyond OEIS sequence A220952

Fischertechnik 15 mm
Fischertechnik 15 mm

Sequence A220952 is best discussed with numbers to base 5. The first 125 values then are:

0,
1,2,3,4,
14,24,34,33,32,31,21,22,23,13,12,11,10,20,30,40,41,42,43,44,
144,244,344,343,342,341,241,242,243,143,142,141,140,240,340,
330,320,310,210,220,230,130,120,110,111,112,113,123,122,121,
131,132,133,233,232,231,221,222,223,213,212,211,311,312,313,
323,322,321,331,332,333,334,324,314,214,224,234,134,124,114,
104,204,304,303,302,301,201,202,203,103,102,101,100,200,300,
400,401,402,403,404,414,424,434,433,432,431,421,422,423,413,
412,411,410,420,430,440,441,442,443,444

In each generation the number of digits is increased by one. The first 25 values form the adjacency condition matrix as defined by Knuth:

04==14==24==34  44
||          ||  ||
03  13==23  33  43
||  ||  ||  ||  ||
02  12  22  32  42
||  ||  ||  ||  ||
01  11  21==31  41
||  ||          ||
00  10==20==30==40
Balsa lumber 8 mm
Balsa lumber 8 mm

Here, the last digit is the y coordinate, and the digit before is the x coordinate. In generation 3, the left-most digit can be interpreted as the z coordinate. Since this is Knuth's original adjacency matrix, and since it shows a letter "n", I named it "Kn".

The curve is space-filling in every dimension

For the cube I wrote a Javascript program with the beautiful framework three.js.

Variants

There is another, trivial matrix:

04==14  24==34  44
||  ||  ||  ||  ||
03  13  23  33  43
||  ||  ||  ||  ||
02  12  22  32  42
||  ||  ||  ||  ||
01  11  21  31  41
||  ||  ||  ||  ||
00  10==20  30==40

A third adjacency matrix shows the structure of the letter "F" and "s", therefore I named it "Fs":

04==14==24==34  44
||          ||  ||
03  13==23==33  43
||  ||          ||
02  12==22==32  42
||          ||  ||
01  11==21==31  41
||  ||          ||
00  10==20==30==40

The corresponding spinning cube shows the coordinate digits.