In orbifold notation the 17 crystallographic groups in the Euclidean plane
("wallpaper groups") are
Instead, we propose the study of the extended Euclidean plane
in secondary school.
At R = [infinity] and equal coefficients (22) the hyperbolic sphere transforms into the Euclidean plane
and Pythagoras' theorem begins to rule again (4).
Let us transform now the definition of the conical curves from the Euclidean plane
to the "spherical plane", i.
By a plane graph we refer to a planar graph together with an embedding in the Euclidean plane
The argument is a theoretical argument showing that the angles in any visible triangle are not "strictly and mathematically" equal to the angles in any Euclidean plane
triangle, but are equal to the angles in a spherical triangle instead.
Given that the disc was open, the orthogonal arcs of the circle were equally open at their two extremities: these arcs were open at their extremities, they were closed by neither a first nor a last point, their terminal points did not belong to them, they belonged -- at the same time as the periphery -- to the complementary part of the space of immersion of the Euclidean plane
Take two fixed points F1 and F2 on Euclidean plane
, then the locus of the points X such that
The following theorem shows that the Lobachevsky plane differs dramatically from the Euclidean plane
from the Ramsey point of view.
0])s and the parabolic surface generated by [alpha] is a Euclidean plane
parallel to the [xi] vector.
That is, in the case of the Euclidean plane
(x, [phi]) wrapping over a cylinder we can identify the azimuthal parameter [phi] with the evolution parameter [tau].
Boroczky (Hungarian Academy of Sciences) builds from the foundation set by Toth (Regular Figures) and Rogers (Packing and Covering) by describing arrangements of congruent convex bodies that either form a packing in a convex container or cover a convex shape, covering arrangements in dimension two (including congruent domains in the Euclidean plane
, translative arrangements, parametric density, and packings and coverings of circular discs) and arrangements in higher dimensions, including packings and coverings by spherical and unit balls, and congruent convex bodies.