A triangle tiling is a tiling of the plane by congruent triangles. Any triangle
tiles the plane (Wells 1991, p. 208).
Let 
be an integer. Divide each side
of an equilateral triangle into 
equal line segments and draw
the corresponding parallel lines to form the triangular
grid illustrated above. The total number 
of triangles of all sizes
and both orientations in this grid is
![N(n)={1/8n(n+2)(2n+1) for n even; 1/8[n(n+2)(2n+1)-1] for n odd.](/images/equations/TriangleTiling/NumberedEquation1.svg) |
(1)
|
The first few values for 
, 2, ... are 1, 5, 13, 27, 48, 78, 118, 170, 235, 315, 411,
525, 658, 812, 988, 1188, 1413, 1665, ... (OEIS A002717).
See also
Equilateral Triangle,
Rectangle Tiling,
Tiling,
Triangle,
Triangle
Counting,
Triangle Packing
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References
Conway, J. H. and Guy, R. K. "How Many Triangles." In The
Book of Numbers. New York: Springer-Verlag, pp. 83-84, 1996.Sloane,
N. J. A. Sequence A002717/M3827
in "The On-Line Encyclopedia of Integer Sequences."Wells,
D. The
Penguin Dictionary of Curious and Interesting Geometry. London, England:
Penguin, pp. 68-69 and 208, 1991.Referenced on Wolfram|Alpha
Triangle Tiling
Cite this as:
Weisstein, Eric W. "Triangle Tiling."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangleTiling.html
Subject classifications
