A triangle dissection divides a triangle into finitely many smaller triangles with nonoverlapping interiors.
Hoggatt and Denman (1961) showed that any obtuse triangle can be divided into eight acute isosceles triangles.
There are 1, 4, 23, 180, 1806, 20198, ... (OEIS A056814) topologically distinct ways to divide a triangle into
,
3, ... smaller triangles (Vicher).
A triangle partition is prime if it does not contain a triangle partition of lower order. The numbers of prime triangle partitions of orders 
, 3, ... are 1, 1, 3, 8, 62, 535, 4213, ... (OEIS A053740).
A specific type of triangle dissection consists of a triangle 
together with an interior point 
such that the original side lengths
and the additional three line segments created by
connecting the triangulation point with the vertices
are all integers. An example of such a dissection is
illustrated above (Pegg).
Other possible dissections allow cut lines to be drawn from arbitrary points along the sides. Allowing only primitive triangles without any parallel lines, isosceles triangles, or similar triangles, the smallest three-piece integer dissection of each of the four possible types is illustrated above.
Similarly, the smallest four-piece integer dissection for each of the 23 possible types is shown above.
Two 5-piece dissections are illustrated above (Pegg).
Laczkovich (1995) proved that every dissection of an equilateral triangle into similar
triangles has a tile angle equal to 60 or 120 degrees.
There exists an infinite family realizing the 120-degree case (E. Pegg Jr.,
pers. comm., Aug. 24, 2026). Let 
be a positive integer,
and let 
be the unique positive real
root of
 |
(1)
|
The polynomial has value 
at 
and tends to 
as 
, so such a root exists.
Uniqueness follows from Descartes' rule of signs.
The law of cosines shows that a triangle
with sides proportional to 
, 
, and 
has an included angle of
120 degrees. Copies of this triangle with scale
factors 
,
,
..., 
,
with the factors 
and 
each occurring twice, form a dissection of an equilateral
triangle into 
similar triangles. Two outer sides
have length proportional to 
, while the third has length proportional to 
. Their equality
follows from
 |
(2)
|
The six cases 
,
..., 6 are shown above in reading order. For even 
, put 
. Then 
is the unique positive real
root of
 |
(3)
|
The first three roots are the tribonacci constant, 
,
and 
,
respectively, since
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
The roots 
are strictly decreasing. Since 
,
![P_(k+1)(x_k)-P_k(x_k)=(x_k-1)x_k^k[x_k^k(x_k^2-1)-1]>0.](/images/equations/TriangleDissection/NumberedEquation4.svg) |
(7)
|
Since 
and 
has only one positive real
root by Descartes' rule of signs, it follows
that 
.
Now 
lies strictly between the two smallest Pisot numbers 
and 
, while 
. It follows that 
is a Pisot number precisely
for 
,
2, or 3. Also, 
follows from 
.
The construction uses only the positive real
root of the displayed polynomial, so the Pisot
property is not required.
