High-Frill Tower

The model Lisa Fonssagrives on the Eiffel Tower wearing a dress by Lucian Lelong, photographed in 1939 by Erwin Blumenfeld for Vogue

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Weight-Botchers

Suppose you have a balance scale and four weights of 1 unit, 2 units, 4 units and 8 units. How many different weights can you match? If you know binary arithmetic, it’s easy to see that you can match any weight up to fifteen units inclusive. With the object in the left-hand pan of the scale and the weights in the right-hand pan, these are the matches:

01 = 1
02 = 2
03 = 2+1
04 = 4
05 = 4+1
06 = 4+2
07 = 4+2+1
08 = 8
09 = 8+1
10 = 8+2
11 = 8+2+1
12 = 8+4
13 = 8+4+1
14 = 8+4+2
15 = 8+4+2+1

Balance scale


The weights that sum to n match the 1s in the digits of n in binary.

01 = 0001 in binary
02 = 0010 = 2
03 = 0011 = 2+1
04 = 0100 = 4
05 = 0101 = 4+1
06 = 0110 = 4+2
07 = 0111 = 4+2+1
08 = 1000 = 8
09 = 1001 = 8+1
10 = 1010 = 8+2
11 = 1011 = 8+2+1
12 = 1100 = 8+4
13 = 1101 = 8+4+1
14 = 1110 = 8+4+2
15 = 1111 = 8+4+2+1

But there’s another set of four weights that will match anything from 1 unit to 40 units. Instead of using powers of 2, you use powers of 3: 1, 3, 9, 27. But how would you match an object weighing 2 units using these weights? Simple. You put the object in the left-hand scale, the 3-weight in the right-hand scale, and then add the 1-weight to the left-hand scale. In other words, 2 = 3-1. Similarly, 5 = 9-3-1, 6 = 9-3 and 7 = 9-3+1. When the power of 3 is positive, it’s in the right-hand pan; when it’s negative, it’s in the left-hand pan.

This system is actually based on base 3 or ternary, which uses three digits, 0, 1 and 2. However, the relationship between ternary numbers and the sums of positive and negative powers of 3 is more complicated than the relationship between binary numbers and sums of purely positive powers of 2. See if you can work out how to derive the sums in the middle from the ternary numbers on the right:

01 = 1 = 1 in ternary
02 = 3-1 = 2
03 = 3 = 10
04 = 3+1 = 11
05 = 9-3-1 = 12
06 = 9-3 = 20
07 = 9-3+1 = 21
08 = 9-1 = 22
09 = 9 = 100
10 = 9+1 = 101
11 = 9+3-1 = 102
12 = 9+3 = 110
13 = 9+3+1 = 111
14 = 27-9-3-1 = 112
15 = 27-9-3 = 120
16 = 27-9-3+1 = 121
17 = 27-9-1 = 122
18 = 27-9 = 200
19 = 27-9+1 = 201
20 = 27-9+3-1 = 202
21 = 27-9+3 = 210
22 = 27-9+3+1 = 211
23 = 27-3-1 = 212
24 = 27-3 = 220
25 = 27-3+1 = 221
26 = 27-1 = 222
27 = 27 = 1000
28 = 27+1 = 1001
29 = 27+3-1 = 1002
30 = 27+3 = 1010
31 = 27+3+1 = 1011
32 = 27+9-3-1 = 1012
33 = 27+9-3 = 1020
34 = 27+9-3+1 = 1021
35 = 27+9-1 = 1022
36 = 27+9 = 1100
37 = 27+9+1 = 1101
38 = 27+9+3-1 = 1102
39 = 27+9+3 = 1110
40 = 27+9+3+1 = 1111

To begin understanding the sums, consider those ternary numbers containing only 1s and 0s, like n = 1011[3], which equals 31 in decimal. The sum of powers is straightforward, because all of them are positive and they’re easy to work out from the digits of n in ternary: 1011 = 1*3^3 + 0*3^2 + 1*3^1 + 1*3^0 = 27+3+1. Now consider n = 222[3] = 26 in decimal. Just as a decimal number consisting entirely of 9s is always 1 less than a power of 10, so a ternary number consisting entirely of 2s is always 1 less than a power of three:

999 = 1000 - 1 = 10^3 - 1 (decimal)
222 = 1000[3] - 1 (ternary) = 26 = 27-1 = 3^3 - 1 (decimal)

If a ternary number contains only 2s and is d digits long, it will be equal to 3^d – 1. But what about numbers containing a mixture of 2s, 1s and 0s? Well, all ternary numbers containing at least one 2 will have a negative power of 3 in the sum. You can work out the sum by using the following algorithm. Suppose the number is five digits long and the rightmost digit is digit #1 and the leftmost is digit #5:

01. i = 1, sum = 0, extra = 0, posi = true.
02. if posi = false, goto step 07.
03. if digit #i = 0, sum = sum + 0.
04. if digit #i = 1, sum = sum + 3^(i-1).
05. if digit #i = 2, sum = sum - 3^(i-1), extra = 3^5, posi = false.
06. goto step 10.
07. if digit #i = 0, sum = sum + 3^(i-1), extra = 0, posi = true.
08. if digit #i = 1, sum = sum - 3^(i-1).
09. if digit #i = 2, sum = sum + 0.
10. i = i+1. if i <= 5, goto step 2.
11. print sum + extra.

As the number of weights grows, the advantages of base 3 get bigger:

With 02 weights, base 3 reaches 04 and base 2 reaches 3: 04-3 = 1.
With 03 weights, base 3 reaches 13 and base 2 reaches 7: 13-7 = 6.
With 04 weights, 000040 - 0015 = 000025
With 05 weights, 000121 - 0031 = 000090
With 06 weights, 000364 - 0063 = 000301
With 07 weights, 001093 - 0127 = 000966
With 08 weights, 003280 - 0255 = 003025
With 09 weights, 009841 - 0511 = 009330
With 10 weights, 029524 - 1023 = 028501
With 11 weights, 088573 - 2047 = 086526
With 12 weights, 265720 - 4095 = 261625...

But what about base 4, or quaternary? With four weights of 1, 4, 16 and 64, representing powers of 4 from 4^0 to 4^3, you should be able to weigh objects from 1 to 85 units using sums of positive and negative powers. In fact, some weights can’t be matched. As you can see below, if n in base 4 contains a 2, it can’t be represented as a sum of positive and negative powers of 4. Nor can certain other numbers:

1 = 1 ← 1
2 has no sum = 2
3 = 4-1 ← 3
4 = 4 ← 10 in base 4
5 = 4+1 ← 11 in base 4
6 has no sum = 12 in base 4
7 has no sum = 13
8 has no sum = 20
9 has no sum = 21
10 has no sum = 22
11 = 16-4-1 ← 23
12 = 16-4 ← 30
13 = 16-4+1 ← 31
14 has no sum = 32
15 = 16-1 ← 33
16 = 16 ← 100
17 = 16+1 ← 101
18 has no sum = 102
19 = 16+4-1 ← 103
20 = 16+4 ← 110
21 = 16+4+1 ← 111
22 has no sum = 112
23 has no sum = 113
24 has no sum = 120
25 has no sum = 121
26 has no sum = 122
27 has no sum = 123
[...]

With a more complicated balance scale, it’s possible to use weights representing powers of base 4 and base 5 (use two pans on each arm of the scale instead of one, placing the extra pan at the midpoint of the arm). But with a standard balance scale, base 3 is the champion. However, there is a way to do slightly better than standard base 3. You do it by botching the weights. Suppose you have four weights of 1, 4, 10 and 28 (representing 1, 3+1, 9+1 and 27+1). There are some weights n you can’t match, but because you can match n-1 and n+1, you know what these unmatchable weights are. Accordingly, while weights of 1, 3, 9 and 27 can measure objects up to 40 units, weights of 1, 4, 10 and 28 can measure objects up to 43 units:

1 = 1 ← 1
2 has no sum = 2
3 = 4-1 ← 10 in base 3
4 = 4 ← 11 in base 3
5 = 4+1 ← 12 in base 3
6 = 10-4 ← 20
7 = 10-4+1 ← 21
8 has no sum = 22
9 = 10-1 ← 100
10 = 10 ← 101
11 = 10+1 ← 102
12 has no sum = 110
13 = 10+4-1 ← 111
14 = 10+4 ← 112
15 = 10+4+1 ← 120
16 has no sum = 121
17 = 28-10-1 ← 122
18 = 28-10 ← 200
19 = 28-10+1 ← 201
20 has no sum = 202
21 = 28-10+4-1 ← 210
22 = 28-10+4 ← 211
23 = 28-4-1 ← 212
24 = 28-4 ← 220
25 = 28-4+1 ← 221
26 has no sum = 222
27 = 28-1 ← 1000
28 = 28 ← 1001
29 = 28+1 ← 1002
30 has no sum = 1010
31 = 28+4-1 ← 1011
32 = 28+4 ← 1012
33 = 28+4+1 ← 1020
34 = 28+10-4 ← 1021
35 = 28+10-4+1 ← 1022
36 has no sum = 1100
37 = 28+10-1 ← 1101
38 = 28+10 ← 1102
39 = 28+10+1 ← 1110
40 = has no sum = 1111*
41 = 28+10+4-1 ← 1112
42 = 28+10+4 ← 1120
43 = 28+10+4+1 ← 1121


*N.B. 40 = 82-28-10-4, i.e. has a sum when another botched weight, 82 = 3^4+1, is used.

WhirlpUlam

Stanislaw Ulam (pronounced OO-lam) was an American mathematician who was doodling one day in 1963 and created what is now called the Ulam spiral. It’s a spiral of integers on a square grid with the prime squares filled in and the composite squares left empty. At the beginning it looks like this (the blue square is the integer 1, with 2 to the east, 3 to the north-east, 4 to the north, 5 to the north-west, 6 to the west, and so on):

Ulam spiral


And here’s an Ulam spiral with more integers:

Ulam spiral at higher resolution


The primes aren’t scattered at random over the spiral: they often fall into lines that are related to what are called polynomial functions, such as n2 + n + 1. To understand polynomial functions better, let’s look at how the Ulam spiral is made. Here is a text version with the primes underlined:


Here’s an animated version:


Here’s the true spiral again with 1 marked as a blue square:

Ulam spiral centred on 1


What happens when you try other numbers at the centre? Here’s 2 at the centre as a purple square, because it’s prime:

Ulam spiral centred on 2


And 3 at the centre, also purple because it’s also prime:

Ulam spiral centred on 3


And 4 at the centre, blue again because 4 = 2^2:

Ulam spiral centred on 4


And 5 at the centre, prime and purple:

Ulam spiral centred on 5


Each time the central number changes, the spiral shifts fractionally. Here’s an animation of the central number shifting from 1 to 41. If you watch, you’ll see patterns remaining stable, then breaking up as the numbers shift towards the center and disappear (the central number is purple if prime, blue if composite):

Ulam whirlpool, or WhirlpUlam


I think the animation looks like a whirlpool or whirlpUlam (prounced whirlpool-am), as numbers spiral towards the centre and disappear. You can see the whirlpUlam more clearly here:
An animated Ulam Spiral pausing at n=11, 17, 41


WhirlpUlam again


Note that something interesting happens when the central number is 41. The spiral is bisected by a long line of prime squares, like this:

Ulam spiral centred on 41


The line is actually a visual representation of something David Wells wrote about in The Penguin Dictionary of Curious and Interesting Numbers (1986):

Euler discovered the excellent and famous formula x2 + x + 41, which gives prime values for x = 0 to 39.

Here are the primes generated by the formula:

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601

You’ll see other lines appear and disappear as the whirlpUlam whirls:

Ulam spiral centred on 17


Primes in line: 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257 (n=0..15)


Ulam spiral centred on 59


Primes in line: 59, 67, 83, 107, 139, 179, 227, 283, 347, 419, 499, 587, 683, 787 (n=0..13)


Ulam spiral centred on 163


Primes in line: 163, 167, 179, 199, 227, 263, 307, 359, 419, 487, 563, 647, 739, 839, 947, 1063, 1187, 1319, 1459, 1607 (n=0..19)


Ulam spiral centred on 233


Primes in line: 233, 241, 257, 281, 313, 353, 401, 457, 521, 593, 673, 761, 857 ((n=0..12)


Ulam spiral centred on 653


Primes in line: 653, 661, 677, 701, 733, 773, 821, 877, 941, 1013, 1093, 1181, 1277, 1381, 1493, 1613, 1741, 1877 (n=0..17)


Ulam spiral centred on 409,333


Primes in line: 409,333, 409337, 409349, 409369, 409397, 409433, 409477, 409529, 409589, 409657, 409733, 409817, 409909, 410009, 410117, 410233 (n=0..15)


Some bisect the centre, some don’t, because you could say that the Ulam spiral has six diagonals, two that bisect the centre (top-left-to-bottom-right and bottom-left-to-top-right) and four that don’t. You could also call them spokes:


If you look at the integers in the spokes, you can see that they’re generated by polynomial functions in which c stands for the central number:

North-west spoke: 1, 5, 17, 37, 65, 101, 145, 197, 257, 325, 401, 485, 577, 677, 785, 901, 1025, 1157, 1297, 1445, 1601, 1765, 1937, 2117, 2305, 2501, 2705, 2917... = c + (2n)^2


South-east spoke: 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625... = c+(2n+1)^2-1


NW-SE diagonal: 1, 5, 9, 17, 25, 37, 49, 65, 81, 101, 121, 145, 169, 197, 225, 257, 289, 325, 361, 401, 441, 485, 529, 577, 625, 677, 729, 785, 841, 901, 961, 1025, 1089, 1157, 1225, 1297, 1369, 1445, 1521, 1601, 1681 = c + n^2 + 1 - (n mod 2)


North-east spoke: 1, 3, 13, 31, 57, 91, 133, 183, 241, 307, 381, 463, 553, 651, 757, 871, 993, 1123, 1261, 1407, 1561, 1723, 1893, 2071... = c + (n+1)^2 - n - 1


South-west spoke: 1, 7, 21, 43, 73, 111, 157, 211, 273, 343, 421, 507, 601, 703, 813, 931, 1057, 1191, 1333, 1483, 1641, 1807, 1981, 2163... = c + (2n)^2 + 2n


SW-NE diagonal: 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, 421, 463, 507, 553, 601, 651, 703, 757, 813, 871, 931, 993, 1057, 1123, 1191, 1261, 1333, 1407, 1483, 1561, 1641... = c + n^2 + n



Elsewhere other-engageable:

All posts interrogating issues around the Ulam spiral