Table 1. Fifteen five-dimensional boundary polytopes of Niggli-reduced cells in .

For a given boundary polytope , the column ‘Condition’ gives the constraints (prior to closure) of the boundary polytope. Boundary polytopes 1 and 2 apply in both the all-acute () and the all-obtuse () branches of the Niggli-reduced cone. Boundary polytopes 8, B, E and F are restricted to the all-obtuse () branch of the Niggli-reduced cone, . Boundary polytopes 6, 7, 9, A, C and D are restricted to the all-acute () branch of . Boundary polytopes 3, 4 and 5 are boundaries of both the all-acute () and the all-obtuse () branches.

Class	Boundary	Condition	Transformation matrix
Equal cell edges	1
	2
90°	3
	4
	5
Face diagonal	6	and
	7	and
	8
	9	and
	A	and
	B
	C	and
	D	and
	E
Body diagonal	F