Why the fifth direction would be small, and the number that is not Planck's
Kaluza's cylinder could be infinitely long. Oskar Klein, in 1926, made it a circle, and the choice carries most of the physics.
Klein asked what a quantum wave would do in Kaluza's five-dimensional world. A wave that goes around a circle must fit a whole number of wavelengths into the circumference, or it cancels itself. So momentum in the fifth direction can only take values in steps: p₅ = nħ/R for a circle of radius R. Since Kaluza's fifth-direction momentum is electric charge, charge comes in whole-number steps. Electric charge is quantized, and a compact extra dimension gives a reason.
The same step shows up as mass. A massless particle moving along the circle looks, from four dimensions, like a massive particle carrying charge. In Klein's framework the mass is of the order of the Planck mass (in Ravndal's conventions, the Planck mass divided by about 23), far heavier than any known particle. The natural size of these states is enormous, and the observed particles are not; that mismatch is an old difficulty of the whole approach.
Klein put in the measured value of the electron charge and the gravitational constant and got a circle of about 0.8 × 10⁻³⁰ cm. He suggested that this smallness, together with the periodicity, might explain why the fifth dimension is never seen in ordinary experiments, because everything averages over it.
That figure is the length of the loop, not its radius. Using Klein's relation as van Dongen prints it, λ = hc√(2κ)/e with κ = 8πG/c⁴, I evaluated it in Gaussian units with current constants and got 8.4 × 10⁻³¹ cm, which reproduces Klein's 0.8 × 10⁻³⁰ cm. Dividing by 2π gives a radius of 1.34 × 10⁻³¹ cm, about 83 Planck lengths (the Planck length is 1.6 × 10⁻³³ cm). Ravndal (arXiv:1309.4113) gets a radius of about 23 Planck lengths, 3.8 × 10⁻³² cm. The two differ by a factor of exactly √(4π) ≈ 3.545. By hand, with the Maxwell term normalised as −¼F² and the coupling κ² = 16πG/c², the charge condition e = κħ/R gives R = 2LP/√α with α = e²/4πħc ≈ 1/137, which is Ravndal's 23 LP. So the factor comes from how charge and field are normalised (Gaussian versus Heaviside–Lorentz). I have not read Klein's original paper to see which normalisation he used. Treat the modern value, about 23 Planck lengths for the radius, as the standard one, and Klein's printed number as the same calculation in older units.
Klein set the extra scalar field (Kaluza's φ) to a constant. Van Dongen notes, citing Thiry, that this is not quite consistent: if you vary the action first and freeze the scalar afterwards, the scalar's own field equation would then require F² = 0, which excludes ordinary electromagnetic fields. Later versions keep the scalar as a dynamical field, the dilaton, and it matters in the string-theory chapters that follow.