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For a plane crystallographic group, the translation subgroup consists of translations by vectors in a rank-two lattice $\Lambda$; the lattice is this additive group of translation vectors. The point group is the image under the linear-part homomorphism $G\to O(2)$, equivalently $G/\Lambda$. Conjugating a translation by an isometry with linear part $A$ gives translation by $Av$, so every point-group element preserves $\Lambda$.

Choose a lattice basis. A point-group rotation is represented in that basis by an integral matrix, because it maps lattice vectors to lattice vectors, and its inverse does likewise. The trace is therefore an integer. In an orthonormal basis the same rotation has trace $2\cos \theta$, which lies between minus two and two. Hence $2\cos \theta \in \{-2,-1,0,1,2\}$. The value two gives the identity, while the other values give angles $\pi$, $\pm 2\pi /3$, $\pm \pi /2$, or $\pm \pi /3$. Thus every nonidentity rotation has orderThis proves the crystallographic restriction theorem from the lattice action rather than assuming the classification of wallpaper groups.

A plane plane crystallographic group is a discrete subgroup of the Euclidean isometry group with compact quotient. Its translations form a normal rank-two Euclidean lattice $\Lambda$; this is the standard translation-lattice structure of a plane crystallographic group. Its point group is the finite group of linear parts, equivalently the quotient by those translations. Conjugating a translation by an isometry sends its translation vector to its image under the linear part, so every point-group element preserves $\Lambda$.

Choose a Euclidean lattice basis. A rotational linear part is then represented by an integer matrix $A$, since it sends the basis to Euclidean lattice vectors. In an orthonormal basis the same map has eigenvalues $e^{i\theta },e^{-i\theta }$, and its trace is independent of basis. ThusThese possibilities give rotation angles, modulo a full turn, $\pi$, $\pm 2\pi /3$, $\pm \pi /2$, $\pm \pi /3$, and $0$, respectively. Consequently the crystallographic restriction theorem isThe restriction comes from integrality of the Euclidean lattice action, not simply from finiteness of the point group.

The point group of a Euclidean crystallographic group is the group of its orthogonal linear parts, equivalently its quotient by the translation subgroup. It preserves the translation Euclidean lattice. In an arbitrary dimension it is a finite orthogonal group; for a plane crystallographic group this lattice preservation restricts the allowed rotation orders.

A plane crystallographic group is a discrete group of Euclidean plane isometries with compact quotient. Its translation subgroup is a rank-two lattice, and the quotient by translations is its finite point group. This separates the periodic translational geometry from the finite rotational and reflection symmetries.