Already in 1882 George FitzGerald and Hendrik Lorentz noted that if there was a contraction in length by a factor Sqrt[1 - v 2 /c 2 ] in any object moving at speed v (with c being the speed of light) then this would explain the result.

If the coordinates along a path are given by an expression s (such as {t, 1 + t, t 2 } ) that depends on a parameter t , and the metric at position p is g[p] , then the length of a path turns out to be Integrate[Sqrt[ ∂ t s . g[s] . ∂ t s], {t, t 1 , t 2 }] and geodesics then correspond to paths that extremize this quantity. In ordinary Euclidean space, such paths are straight lines, so that the length of a path between points with lists of coordinates a and b is just the ordinary Euclidean distance Sqrt[(a - b) .

In addition: • GoldenRatio is the solution to x  1 + 1/x or x 2  x + 1 • The right-hand rectangle in is similar to the whole rectangle when the aspect ratio is GoldenRatio • Cos[ π /5]  Cos[36 ° ]  GoldenRatio/2 • The ratio of the length of the diagonal to the length of a side in a regular pentagon is GoldenRatio • The corners of an icosahedron are at coordinates Flatten[Array[NestList[RotateRight, {0, (-1) #1 GoldenRatio, (-1) #2 }, 3]&, {2, 2}], 2] • 1 + FixedPoint[N[1/(1 + #), k] &, 1] approximates GoldenRatio to k digits, as does FixedPoint[N[Sqrt[1 + #],k]&, 1] • A successive angle difference of GoldenRatio radians yields points maximally separated around a circle (see page 1006 ).

The first 2 m elements in the sequence can be obtained from (see page 1081 ) (CoefficientList[Product[1 - z 2 s , {s, 0, m - 1}], z] + 1)/2 The first n elements can also be obtained from (see page 1092 ) Mod[CoefficientList[Series[(1 + Sqrt[(1 - 3x)/(1 + x)])/ (2(1 + x)), {x, 0, n - 1}], x], 2] The sequence occurs many times in this book; it can for example be derived from a column of values in the rule 150 cellular automaton pattern discussed on page 885 .

The total number of ways that integers less than n can be expressed as a sum of d squares is equal to the number of integer lattice points that lie inside a sphere of radius Sqrt[n] in d -dimensional space.

The overall maximum gap occurs at c = 1/2 Sqrt[5 - √ 17 ]  .

In general, the radius of a circle inscribed between three other touching circles that have radii p , q , r is p q r/(p q + p r + q r + 2 Sqrt[p q r (p + q + r)]) .)

For large j i they are approximated by Cos[ θ + π /4]/Sqrt[12 π v] , where v is the volume of the tetrahedron and θ is a deficit angle.