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Ruler and compass constructions in the Lemniscate and the 17-gon

Mariángeles Gómez-Molleda, Joan-C. Lario

Abstract

We present several ruler and compass practical geometric constructions that can be performed in the lemniscate curve. To be precise, we provide recipes for halving, doubling, adding, subtracting, and transferring lemniscate arcs with ruler and compass. This note complements the instructions for the lemnatomic equilateral triangle and pentagon discussed in \cite{GoLa}, giving the details for the construction of the lemnatomic regular $17$-gon.

Ruler and compass constructions in the Lemniscate and the 17-gon

Abstract

We present several ruler and compass practical geometric constructions that can be performed in the lemniscate curve. To be precise, we provide recipes for halving, doubling, adding, subtracting, and transferring lemniscate arcs with ruler and compass. This note complements the instructions for the lemnatomic equilateral triangle and pentagon discussed in \cite{GoLa}, giving the details for the construction of the lemnatomic regular -gon.

Paper Structure

This paper contains 9 sections, 1 theorem, 31 equations, 14 figures.

Table of Contents

  1. Introduction.
  2. Auxiliary constructions
  3. Halving arcs.
  4. Doubling arcs.
  5. Addition and subtraction of arcs.
  6. Other constructions.
  7. The 17-gon
  8. Radical expressions for the radii of the lemniscate 17-gon.
  9. Construction of the 17-gon by ruler and compass.

Key Result

Theorem 1

The lemniscate can be divided into $n$ equal parts with ruler and compass if $n=2^ap_1 p_2 \dots p_m,$ where $p_1, \dots, p_m$ are distinct Fermat primes,

Figures (14)

  • Figure 1: The lemniscate arc-length function
  • Figure 2: Negative of the solutions of $x^2+2\,x+1/2=0$
  • Figure 3: Halving the lemniscate arc determined by $u_\varphi$
  • Figure 4: Step 1: Build the point $A$. The segment $OA = \sec(2\varphi)+\tan(2\varphi)$
  • Figure 5: Step 2: Build the point $B$. The segment $OB = \cos(2\theta)$
  • ...and 9 more figures

Theorems & Definitions (2)

  • Theorem : Abel, 1827
  • Remark 1