Geometry

Euler’s classification (1776) of isometries in dimensions ≤ 3.

Euler’s classification (1776)
of isometries in dimensions ≤ 3

Isometry Eucl	Fixed Point?
	Yes	No
OP	rotations ρ (dim ≥ 2)	translations τ (dim ≥ 1) screw (dim ≥ 3)
OR	reflections σ (dim ≥ 1) rotary reflections (dim ≥ 3)	glide reflections γ (dim ≥ 2)

Notes:

1. OP = orientation-preserving = even = direct = Isom⁺
   OR = orientation-reversing = odd =opposite = Isom^-


2. Each non-identity isometry is in precisely one of the above categories.


3. Wikipedia discusses:
   Classification of Euclidean plane isometries and
   Overview of isometries in up to three dimensions.


4. Reflections
   More precisely, we mean reflections in a hyperplane of codimension 1
   (thus in 2-space, in a line, in 3-space, in a plane)
   with respect to the orthogonal line to the hyperplane.


5. Rotations in 2-space are with respect to a point,
   in 3-space with respect to a line.
   Thus rotations are always within a plane,
   about the orthogonal complement.
   Of course, in dimensions ≥ 4 things get more complicated,
   and are best discussed in terms of
   the decomposition of orthogonal operators studied in linear algebra.
   This link, if it is still active, gives the relevant theorem.
   There is also Wikipedia’s discussion of the orthogonal group,
   but that covers much more than is immediately relevant.


6. Glide reflections:
   The direction of the glide is always within the reflecting hyperplane:
   in 2-space the reflecting line, in 3-space the reflecting plane.
   Glide reflections in 3-space are sometimes called
   “glide plane operations”.


7. Screw = rotary translation = glide rotation


8. Inversions
   By inversion
   we mean what is sometimes called
   “central inversion” or “inversion in a point.”
   In 2-space, an inversion through a point
   is just a half-turn about the point, thus a rotation.
   In 3-space, for inversion through a point
   we may choose any line through that point,
   perform a half-turn rotation around that line
   combined with a reflection in
   the plane orthogonal to that line and through that point,
   thus we have a rotary reflection.
   This example shows that although each isometry
   is in one and only one of the above categories,
   its precise expression within that category is not unique.


9. The familiar (from high school geometry) property of
   congruence of figures in the plane
   simply means that
   one figure is the image of the other under an isometry.


10. There is doubt as to how much of that classification
    Euler really discovered.
    I’m giving him the benefit of the doubt,
    following Coolidge’s History of Geometrical Methods
    as cited in George Martin’s Transformation Geometry.