Let
Δ
A
B
C
\Delta ABC
be a triangle with vertices A, B, and C. It is "bisected" as follows: choose a/the longest side (say AB) of
Δ
A
B
C
\Delta ABC
, let D be the midpoint of AB, then replace
Δ
A
B
C
\Delta ABC
by two triangles
Δ
A
D
C
\Delta ADC
and
Δ
D
B
C
\Delta DBC
. Let
Δ
01
{\Delta _{01}}
be a given triangle. Bisect
Δ
01
{\Delta _{01}}
into two triangles
Δ
11
{\Delta _{11}}
and
Δ
12
{\Delta _{12}}
. Next bisect each
Δ
1
i
,
i
=
1
,
2
{\Delta _{1i}},\;i = 1,2
, forming four new triangles
Δ
2
i
,
i
=
1
,
2
,
3
,
4
{\Delta _{2i}},\;i = 1,2,3,4
. Continue thus, forming an infinite sequence
T
j
,
j
=
0
,
1
,
2
,
…
{T_j},\;j = 0,1,2, \ldots
, of sets of triangles, where
T
j
=
{
Δ
j
i
:
1
⩽
i
⩽
2
j
}
{T_j} = \left \{ {{\Delta _{ji}}:1 \leqslant i \leqslant {2^j}} \right \}
. Let
m
j
{m_j}
denote the mesh of
T
j
{T_j}
. It is shown that there exists
N
=
N
(
Δ
01
)
N = N({\Delta _{01}})
such that, for
j
⩾
N
j \geqslant N
,
m
2
j
⩽
(
3
/
2
)
N
(
1
/
2
)
j
−
N
m
0
{m_{2j}} \leqslant {(\sqrt 3 /2)^N}{(1/2)^{j - N}}{m_0}
, thus greatly improving the previous best known bound of
m
2
j
⩽
(
3
/
2
)
j
m
0
{m_{2j}} \leqslant {(\sqrt 3 /2)^j}{m_0}
. It is also shown that only a finite number of distinct shapes occur among the triangles produced, and that, as the method proceeds,
Δ
01
{\Delta _{01}}
tends to become covered by triangles which are approximately equilateral in a certain sense.