REP-TILES
In 1962, Solomon Golomb began exploring shapes that could be used to create larger and
smaller copies of themselves. He named these shapes “replicating figures” or “rep-tiles” and
thus began an interesting geometric study that builds on the idea of similarity.
Rep-tiles are related to tessellations (tilings), but they are different in one important factor.
Rep-tiles are a subset of tessellations—that is, all rep-tiles are tessellations but not all
tessellations are rep-tiles. For example, every triangle both tessellates and is a rep-tile because
copies of a triangle can be combined to make a larger, similar copy of the same triangle (See
below). On the other hand, the regular hexagon tessellates but is not a rep-tile because no
number of tessellating hexagons will ever create a larger hexagon (see below).
Example Rep-tile
Not a Rep-tile
Another example of a rep-tile is the pentagon (known as the Sphinx) that is shown below. Using
four copies of the same size, another Sphinx can be created. Because four copies can be used
to create this rep-tile, it is called a rep-4 polygon.
Sphinx
Rep-tiles
J. Wanko & G. Hawk – Miami University
Sphinx rep-4
It Figures - 13 NCTM 2011 Annual Meeting Each of the shapes below is also a rep-4 polygon. Can you find a way to fit together four copies
of a shape to make a larger shape that is mathematically similar to the original shape? Can you
find a way to fit together some other number of copies (n) to make a larger similar shape (to
show that a shape is also rep-n)?
Small L
Large L
Right Trapezoid
Isosceles Trapezoid
P Pentomino
14 - It Figures
NCTM 2011 Annual Meeting
Rep-tiles J. Wanko & G. Hawk – Miami University Students might find it helpful to have a frame in which they could place copies of a shape to
explore rep-tiles. Here are frames for the rep-4 explorations of the shapes from the previous
page.
Rep-tiles
J. Wanko & G. Hawk – Miami University
It Figures - 15 NCTM 2011 Annual Meeting 16 - It Figures
NCTM 2011 Annual Meeting
Rep-tiles J. Wanko & G. Hawk – Miami University