Interlocking floor tiles.
Version of Wednesday 16 August 2017.
Dave Barber's other pages.

This report describes a modular system of floor tile design, intended particularly for interlocking pieces formed from cushion-like material. The user arranges multiple tiles on a hard floor to provide a soft surface for play, exercise or other needs. Because the tiles are not permanently attached to one another, that they can be rearranged from time to time.

First is covered a rectangular system, then a hexagonal.

Rectangular.

§1. Figure 1A shows the shape of the fundamental module.

figure 1A figure 1B  The projections on the top and bottom of this tile are keys, and the recesses on the left and right are locks. Without keys or locks, 1A would become the ordinary square of figure 1B.

The exact size and shape of the keys and locks is not specified in this report, because that aspect of the design would depend on the material of which the tiles are made and the purpose for which they would be used.

Likely choices for "one unit" would be lengths in the vicinity of 3 inches or 8 centimeters. The typical tile, however, will not be a single module, which is inconveniently small for most flooring purposes. Rather, most tiles will be multiples of the module, for example a rectangular arrangement 6 units by 4 units.

§2. Figure 2 shows how modules connect, in rather the manner of a jigsaw puzzle.

figure 2 §3. Figure 3 shows some shapes, differing in the arrangement of keys and locks, which had been considered for this plan but which ultimately were not chosen. Note that tiles such as 3B are at risk of mechanical failure due to the thin isthmus (see red circle).

figure 3 If all modules are to be based on the same pattern, then each will need two keys and two locks, such as figure 1A or 3C. Even if the isthmus problem can be resolved, one reason to reject 3C is that tiles cannot be rotated; by contrast a 1A tile can be rotated 180 degrees. Rotation may be desired if the tile is printed with an asymmetrical decorative pattern. A second reason is that, using the 3C module, edge pieces (introduced in figure 5 below) will have to be produced in twice as many varieties as with 1A.

A two-module system, with 3A and 3B alternating, would also be effective. However, this raises the question of why two modules should be used if one will suffice.

§4. Figure 4 displays a tile of more typical size, although drawn at a smaller scale than the tiles above. The dashed-dotted lines, which would not appear on the actual manufactured product, make it conspicuous how many modules are included. (Some computers have difficulty displaying the fine lines; this is related to aliasing and dithering.)

figure 4 §5. Figure 5 is a multiple-tile arrangement. Each edge tile has one flat side: no keys or locks. Similarly, each corner tile has two flat sides. As can be seen, this modular system does not restrict the tiles to being laid in a checkerboard pattern.

figure 5 Because edge and corner tiles are of essentially the same design and sizes as interior tiles, calculation of the number of tiles required to cover a given area is simplified.

§6. The catalogue of possible sizes and shapes of tiles can grow huge, particularly when edge and corner pieces are provided. Because this could burden manufacturers and confuse customers, a practical response is to stipulate the following:

• every tile must have a shape that is rectangular overall — no "L" or "T" shapes for instance;
• every tile must have a width and a length that are even numbers of units;
• every edge tile must measure two units in the direction perpendicular to the flat side;
• every corner tile, because it has two flat sides, must measure two units in both directions.
Even with these constraints, a checkerboard pattern is not inevitable, as shown in figure 6A.

figure 6A Note the following contrast pertinent to this multiple-of-two system.

 figure 6Bcompatible figure 6Cincompatible  Some manufacturers might produce tiles that can be flipped over, which would make 6B and 6C the same tile viewed from opposite sides.

A further limitation, confining tiles to multiples of four units, can still give convenient results:

• with unit size 3 inches, each tile would have dimensions that are a multiple of 1 foot.
• with unit size 6.25 centimeters, each tile would have dimensions that are a multiple of one-fourth of a meter;
• with unit size 8.33 centimeters, each tile would have dimensions that are a multiple of one-third of a meter.

§7. The overall layout can have an inside corner — no special tiles are required, as in figure 7.

figure 7 §8. Figure 8 is a modification of figure 6. Edge and corner tiles are now one unit wide, while interior pieces are unchanged.

figure 8 §9. Sufficient for many purposes would be the following selection of ten tiles from the multiple-of-two system.

figure 9 §10. If the unit size is cut in half, then figure 10 below, which is from the multiple-of-four system, becomes equivalent to figure 9.

figure 10 Hexagonal.

§11. Figure 11A shows the shape of the fundamental hexagonal module, with keys and locks similar to those of the rectangular tiles above.

figure 11  Without keys or locks, 11A would become the regular hexagon of figure 11B.

With hexagonal modules, likely choices for "one unit" would be lengths in the vicinity of 12 inches or 25 centimeters, and the typical tile would be a single module. This is in contrast to the rectangular tiles discussed above, where one tile usually includes multiple modules.

§12. Figure 12 shows how tiles are assembled into an ordinary hexagonal grid.

figure 12 §13. Figure 13 shows some tiles composed of multiple hexagons. Although feasible, they are not developed further in this report.

figure 13   §14A. As with the rectangular system, edge pieces should be provided. There is much to be said.

Type A is a full hexagonal tile which is modified by having one or two sides flattened by removing keys and locks.

type A  In a typical tile arrangement, there would rarely be need for two non-adjacent sides of one tile to be flattened. Similarly infrequent would be the call to flatten at least one, but not all, of the locks and keys on any one side of a tile.

A wealth of additional edge tiles can be produced by cutting, resulting in a tile that is something less than a regular hexagon; a variety are displayed below starting in §14B. They make it possible for a large assembly to have straight edges, as shown starting in §15.

As there are many ways to cut a tile, producers will no doubt offer only a limited range of factory-cut pieces. Of mitigation is that these floor tiles are often made of materials that can be cut with an ordinary knife, allowing the customer to trim as desired. To aid, the underside of each tile can be marked with guide lines, as below.

underside markings   The examples of cut tiles presented here are organized into types; other taxonomies are certainly possible. In this system, a cut parallel to some side of the original hexagon (i.e. a hex side) is a B-cut; or perpendicular, a C-cut. Cuts in other directions are not addressed.

• type A — no cuts, as above
• type B — one B-cut
• type C — one C-cut
• type D — two B-cuts, oblique and intersecting
• type E — two C-cuts, oblique and intersecting
• type F — one B-cut and one C-cut, perpendicular and intersecting, the C-cut farther clockwise than the B-cut
• type G — one B-cut and one C-cut, perpendicular and intersecting, the B-cut farther clockwise than the C-cut
• type H — one B-cut and one C-cut, oblique and intersecting

It turns out that the angles between the following are multiples of 30 degrees:

• any two hex sides;
• any cut side and any hex side;
• any two cut sides.

If a tile with one cut side is to serve as a corner of the overall layout, it is often useful to flatten one of the hex sides next to the cut side. On the other hand, if a tile already has two cut sides, flattening of a hex side will rarely be beneficial.

Examples of cut tiles appear below, each with a fractional measure of its area relative to a full hexagon. The least common denominator of all the fractions is 144, as suggested by the diagram below, which contains 24 equilateral triangles each holding 6 right triangles.

144 = 24 × 6 §14B. In B2 and B3 below, the 60-degree angle between the cut side and each adjoining side, being acute, makes it unlikely that there would be an occasion to flatten those adjoining sides.

type B1: 114144 = 1924   type B2: 72144 = 12 type B3: 30144 = 524  §14C. In C3 below, the highly acute 30-degree angle leads to a severe isthmus problem (see red circle), meaning that the piece is probably not feasible.

type C1: 120144 = 56   type C2: 72144 = 12 type C3: 24144 = 16    §14D. Each tile among the D types below has two B-cuts.

type D11: 96144 = 23 type D22: 48144 = 13 type D33: 12144 = 112
two cuts like B1 two cuts like B2 two cuts like B3   types D12 and D21: 66144 = 1124 types D23 and D32: 24144 = 16
one cut like B1, one cut like B2 one cut like B2, one cut like B3    Although D21's cuts are the reverse of D12's, the keys and locks are not reversed (see red circles) — hence D21 is not exactly a mirror image of D12. The same applies to D23-D32 and many other pairs.

§14E. Each tile among the E types has two cuts in the manner of C1 or C2. Any C3-style cuts would result in a piece that is probably not feasible due to the 30-degree angle.

type E11: 104144 = 1318 type E22: 48144 = 13 types E12 and E21: 68144 = 1736
two cuts like C1 two cuts like C2 one cut like C1, one cut like C2    §§14F&G. The cuts of type F can be reversed to produce type G. Within each tile, the cuts are perpendicular.

types F11 and G11: 93144 = 3148 types F12 and G12: 57144 = 1948
one cut like B1, one cut like C1 one cut like B1, one cut like C2    types F21 and G21: 60144 = 512 types F22 and G22: 36144 = 14
one cut like B2, one cut like C1 one cut like B2, one cut like C2    types F31 and G31: 27144 = 948 types F32 and G32: 15144 = 548
one cut like B3, one cut like C1 one cut like B3, one cut like C2    F13 and G13 below are too small for most purposes, but they serve to reveal an asymmetry: G13 has the isthmus problem of C3, but F13 does not.

types F13 and G13: 21144 = 748
one cut like B1, one cut like C3  §14H. Within each tile, the cuts are oblique.

types H1 and H2: 108144 = 34 types He and H4: 60144 = 512
one cut like B1, one cut like C1 one cut like B2, one cut like C2    §14I. A manufacturer seeking a practical subset of edge pieces might opt for the tiles listed in column 0 in the table below, although column 2 is a reasonable alternative. However, tiles from column 1 will generally necessitate inclusion of tiles from both of columns 0 and 2.

number of sides that are half-size
012
A1, A2
B2B1, B3
C1, C3C2
D22D12, D21, D23, D32D11, D33
E11E12, E21E22
F21F11, F22, F31F12, F32
G21G11, G22, G31G12, G32
H1, H2, H3, H4

§15. Figures 15A, 15B, and 15C are the same except for where the edges have been cut.

figure 15A figure 15B figure 15C §16. Even though tiles are hexagonal, an arrangement of rectangular outline is possible:

figure 16A figure 16B §17. For a decorative effect, edge tiles can be used in the interior of an arrangement, although the lack of locks and keys at junctions increases the risk that tiles will be accidentally dislodged. With tiles of sufficient thickness, however, one can be glued to another if a permanent connection is acceptable.

figure 17A The length of the cut side of tile B2 equals twice the length of a hex side, meaning that four sets of locks and keys will precisely fit the cut side, yielding tile X2. Meanwhile, each cut side of a D22 equals a hex side in length, whence tile Z22. Together, they suggest tile Y1. With tiles B1 and B3, three sets of locks and keys fit on the cut side.

Many of these tiles have an acute angle (60 degrees) and may suffer from the isthmus problem depending on the details of the lock-and-key design.

figure 17B1    compare B2 compare B1 and B3 compare D22 compare D11

figure 17B2  The cut sides of C2 and E22 are not multiples of a hex side, so locks and keys cannot be expected to work well.

These tiles can be assembled in a manner that obfuscates any hexagonal grid:

figure 17C 