OEIS/FASS curves

Steel wire

Beyond OEIS sequence A220952

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

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 cube can be viewed:

• as a colored line with coordinates
• as a solid 3D model
• by downloading the STL file and uploading it to the free service of viewstl.com