[table][tr][td]One of the many celtic motifs is familiar with four Yin Yang symbols.[br]Drawing this symbol with compass and ruler is fun, but there is also algebra involved.[br]There are only two parameters: the size of the circle and the width of the band.[br]Say OA=1, the width of the band AE = AM is limited. It can be almost zero, but it is always less than one.[/td][td][/td][/tr][/table]

Draw this celtic symbol with compass and ruler.

Key characteristic of this motif is the width of the black and the white band.[br]Point [math]B[/math] is at line [math]AO[/math]. The closer point [math]B[/math] is to point [math]A[/math] the larger the drops and the smaller the width of the band. The closer to point [math]O[/math], the smaller the drops and the larger the width of the band.[br]Say [math]\text{AO = 1}[/math]. Proof that:[br][list][*]the exact values of the limiting values of the position of point [math]B[/math] on line [math]AO[/math] are [math]OB=\sqrt{2}-1[/math] and [math]OB=2-\sqrt{2}[/math]. [/*][*]the maximum width of the white and the black band is [math]AE=AM=2-\sqrt{2}[/math] [/*][/list]

Say that [math]AO=1[/math] and that the radius of the greater circle is [math]AB=a[/math] and that the radius of the smaller circle is [math]BE=b[/math] and that the width of the band is [math]AE=AM=a=b=c[/math]. Proof that[br][list][*] [math]a=\sqrt{2}-1+\left(1-\frac{1}{2}\sqrt{2}\right)\cdot c[/math] [/*][*] [math]b=\sqrt{2}-1-\frac{1}{2}\sqrt{2}\cdot c[/math][/*][/list]