OEIS/FASS curves: Difference between revisions
Jump to navigation
Jump to search
imported>Gfis No edit summary |
imported>Gfis No edit summary |
||
| Line 1: | Line 1: | ||
[[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, | ||
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, | |||
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, | ||
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 | ||
| Line 19: | Line 23: | ||
|| || || | || || || | ||
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
Beyond OEIS sequence A220952
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
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.