There are (at least) two mathematical constants associated with Theodorus. The first Theodorus's constant is the elementary algebraic
number,
i.e., the square root of 3. It has decimal expansion
Another constant sometimes known as the constant of Theodorus is the slope of a continuous analog of the discrete Theodorus
spiral due to Davis (1993) at the point , given by
Davis, P. J. Spirals from Theodorus to Chaos. Wellesley, MA: A K Peters, 1993.Finch,
S. R. Mathematical
Constants II. Cambridge, England: Cambridge University Press, 2019.Gautschi,
W. "The Spiral of Theodorus, Special Functions, and Numerical Analysis."
In Spirals
from Theodorus to Chaos (Ed. P. J. Davis). Wellesley, MA: A K Peters,
pp. 67-87, 1993.Jones, M. F. "Approximations to the Square
Roots of the Primes Less Than 100." Math. Comput.22, 234-235,
1968.Sloane, N. J. A. Sequences A002194/M4326,
A004547, A040001,
and A226317 in "The On-Line Encyclopedia
of Integer Sequences."Uhler, H. S. "Approximations Exceeding
Decimals for ,
,
,
and Distribution of Digits in Them." Proc. Nat. Acad. Sci. USA37,
443-447, 1951.Wells, D. The
Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, pp. 34-35, 1986.
,
i.e., the square root of 3. It has decimal expansion
.
has continued fraction [1, 1, 2, 1, 2, 1, 2,
...] (OEIS A040001). In binary, it is represented
by
, given by
![1/2-sum_(k=1)^(infty)(-1)^k[zeta(k+1/2)-1]](/images/equations/TheodorussConstant/Inline13.svg)
is the Riemann zeta function.
Decimals for 
,
,
,
and Distribution of Digits in Them." Proc. Nat. Acad. Sci. USA 37,
443-447, 1951.Wells, D.