All fans of recreational math love palindromic numbers. It’s mandatory, man. 101, 727, 532235, 8810188, 1367755971795577631 — I love ’em! But where can you go after palindromes? Well, you can go to palindromes in a higher dimension. Numbers like 101, 727, 532235 and 8810188 are 1-d palindromes. That is, they’re palindromic in one dimension: backwards and forwards. But numbers like 181818189 and 646464640 aren’t palindromic in one dimension. They’re palindromic in *two* dimensions:

1 8 1

8 9 8

1 8 1

`
`n=181818189

6 4 6

4 0 4

6 4 6

n=646464640

They’re 2-d palindromes or spiral numbers, that is, numbers that are symmetrical when written as a spiral. You start with the first digit on the top left, then spiral inwards to the center, like this for a 9-digit spiral (9 = 3×3):

And this for a 36-digit spiral (36 = 6×6):

Spiral numbers are easy to construct, because you can reflect and rotate the numbers in one triangular slice of the spiral to find all the others:

↓

↓

You could say that the seed for the spiral number above is 7591310652, because you can write that number in descending lines, left-to-right, as a triangle.

Here are some palindromic numbers with nine digits in base 3 — as you can see, some are both palindromic numbers and spiral numbers. That is, some are palindromic in both one and two dimensions:

1 0 1

`
`0 1 0

1 0 1

n=101010101

1 0 1

0 2 0

1 0 1

n=101010102

1 1 1

1 0 1

1 1 1

n=111111110

1 1 1

1 1 1

1 1 1

n=111111111

2 0 2

0 1 0

2 0 2

n=202020201

2 0 2

0 2 0

2 0 2

n=202020202

2 2 2

2 1 2

2 2 2

n=222222221

2 2 2

2 2 2

2 2 2

n=222222222

But palindromic primes are even better than ordinary palindromes. Here are a few 1-d palindromic primes in base 10:

101

151

73037

7935397

97356765379

1091544334334451901

1367755971795577631

70707270707

39859395893

9212129

7436347

166000661

313

929

And after 1-d palindromic primes, you can go to 2-d palindromic primes. That is, to spiral primes or sprimes — primes that are symmetrical when written as a spiral:

3 6 3

6 7 6

3 6 3

`
`n=363636367 (prime)

seed=367 (see definition above)

9 1 9

1 3 1

9 1 9

n=919191913 (prime)

seed=913

3 7 8 6 3 6 8 7 3

7 9 1 8 9 8 1 9 7

8 1 9 0 9 0 9 1 8

6 8 0 5 5 5 0 8 6

3 9 9 5 7 5 9 9 3

6 8 0 5 5 5 0 8 6

8 1 9 0 9 0 9 1 8

7 9 1 8 9 8 1 9 7

3 7 8 6 3 6 8 7 3

n=378636873786368737863687378636879189819189819189819189819090909090909090555555557 (prime)

seed=378639189909557 (l=15)

And why stop with spiral numbers — and sprimes — in two dimensions? 363636367 is a 2-sprime, being palindromic in two dimensions. But the digits of a number could be written to form a symmetrical cube in three, four, five and more dimensions. So I assume that there are 3-sprimes, 4-sprimes, 5-sprimes and more out there. Watch this space.