The x^x Spindle

threads

1

3

5

7

9

11

13

15

color scheme

GrayTones

It is easy to sketch the graph of

y=

x

for

x>0

. For

x<0

,

x

is only defined for rational

x

with odd denominator, unless complex values are allowed. In this case, set

x

=

x(log(x)+2πk)



, where log represents the principal branch of the complex logarithm; other branches differ by an integer multiple of

2π

and these may be specified using the parameter

k

in the formula. Thus, any

k

value determines a 3D graph where each

x

value determines the point in space

(x,re(

x

),im(

x

))

. The collection of all of these graphs, or threads, forms the

x

spindle.