Lovebirds Tilings

Introduction

This formula holds for every number k:

Σ(i=1; k) i3 = T(k)2,

where T(k) is the kth triangular number. This formula implies that a square with side T(k) might be dissected into 1 square of side 1, 2 squares of side 2, and so on up to k squares of side k. Such a tiling is a partridge tiling, named after the partridge in the song The Twelve Days of Christmas. It was first proposed and discovered by Robert Wainwright. See the August 2002 issue of Erich Friedman's Math Magic for a survey of known partridge tilings.

In 2021 I introduced lovebirds tilings. These use a double set of copies of a shape, with scale factors 1…k, to construct two copies of the shape at scale T(k).

Any plane shape with a partridge tiling can use two copies of the tiling to form a lovebirds tiling. Here I show polyforms that can form lovebirds tilings but not partridge tilings, or can form lovebirds tilings with fewer sizes of tiles than their smallest known partridge tilings. If you find a new lovebirds tiling with this property, please write.

Polyominoes

The monomino, or square, has a minimal partridge tiling with k=8. It has a minimal lovebirds tiling with k=7:

The domino has partridge number 7. It has lovebirds number 3:

It follows that any parallelogram with sides in the ratio 1:2 has a lovebirds number of 3 or less. This includes the straight tetriamond.

The I tetromino has partridge number 7. Its lovebirds number is 6:

The L tetromino has no known partridge tiling. Its lovebirds number is 3:

The P hexomino has no known partridge tiling. Its lovebirds number is 3:

The 2×3 rectangle has partridge number 7. It has lovebirds number 3:

The 3×4 rectangle has partridge number 7. It has lovebirds number 3:

Polyiamonds

The moniamond has partridge number 9. Its lovebirds number is 7:

It follows that any triangle has lovebirds number 7 or less.

Polyaboloes

The monabolo has partridge number 8. Erich Friedman discovered that it has lovebirds number 3:

From this Erich deduced that the lovebirds number of any right triangle is at most 3.

Polydoms

The monodom has partridge number 6. Its lovebirds number is 3:

This tridom has partridge number 6. Its lovebirds number is 4:

Polydrafters

The monodrafter has partridge number 4. Its lovebirds number is 3:

This tridrafter has partridge number 6 and lovebirds number 3:

Last revised 2021-04-07.


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Col. George Sicherman [ HOME | MAIL ]