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Miller indices of a plane (hkl) and a direction [hkl].The intercepts on the axes are at a/ h, b/ k and c/ l. The International Union of Crystallography (IUCr) gives the following definition: "The law of rational indices states that the intercepts, OP, OQ, OR, of the natural faces of a crystal form with the unit-cell axes a, b, c are inversely proportional to prime integers, h, k, l.
Snell's law (also known as the Snell–Descartes law, the ibn-Sahl law, [1] and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air.
For equal permeabilities (e.g., non-magnetic media), if θ i and θ t are complementary, we can substitute sin θ t for cos θ i, and sin θ i for cos θ t, so that the numerator in equation becomes n 2 sin θ t − n 1 sin θ i, which is zero (by Snell's law).
So, substituting these values into the formula, ... The laws result from canceling exponentials and the appropriate law of indices. Starting with the first law:
A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens. If the refractive index of a medium is not constant but varies gradually with the position, the material is known as a gradient-index (GRIN) medium and is described by gradient index ...
In mathematics, a law is a formula that is always true within a given context. [1] Laws describe a relationship , between two or more expressions or terms (which may contain variables ), usually using equality or inequality , [ 2 ] or between formulas themselves, for instance, in mathematical logic .
Haüy's method is named the law of decrements. [11]: 322 The law of rational indices was not stated in its modern form by Haüy, but it is directly implied by his law of decrements. [11]: 333 Cube to octahedron by truncation of all vertices (holohedry) Cube to tetrahedron by truncation of alternate vertices (hemihedry)
A subgroup H of finite index in a group G (finite or infinite) always contains a normal subgroup N (of G), also of finite index. In fact, if H has index n, then the index of N will be some divisor of n! and a multiple of n; indeed, N can be taken to be the kernel of the natural homomorphism from G to the permutation group of the left (or right ...