We present a procedure for bisecting a tetrahedron T successively into an infinite sequence of tetrahedral meshes
T
0
,
T
1
,
T
2
,
…
{\mathcal {T}^0},{\mathcal {T}^1},{\mathcal {T}^2}, \ldots
, which has the following properties: (1) Each mesh
T
n
{\mathcal {T}^n}
is conforming. (2) There are a finite number of classes of similar tetrahedra in all the
T
n
,
n
≥
0
{\mathcal {T}^n},n \geq 0
. (3) For any tetrahedron
T
i
n
{\mathbf {T}}_i^n
in
T
n
,
η
(
T
i
n
)
≥
c
1
η
(
T
)
{\mathcal {T}^n},\eta ({\mathbf {T}}_i^n) \geq {c_1}\eta ({\mathbf {T}})
, where
η
\eta
is a tetrahedron shape measure and
c
1
{c_1}
is a constant. (4)
δ
(
T
i
n
)
≤
c
2
(
1
/
2
)
n
/
3
δ
(
T
)
\delta ({\mathbf {T}}_i^n) \leq {c_2}{(1/2)^{n/3}}\delta ({\mathbf {T}})
, where
δ
(
T
′
)
\delta ({\mathbf {T’}})
denotes the diameter of tetrahedron
T
′
{\mathbf {T’}}
and
c
2
{c_2}
is a constant. Estimates of
c
1
{c_1}
and
c
2
{c_2}
are provided. Properties (2) and (3) extend similar results of Stynes and Adler, and of Rosenberg and Stenger, respectively, for the 2-D case. The diameter bound in property (4) is better than one given by Kearfott.