Mathematics can be a powerful tool to create works of art. By using mathematical concepts, we can draw interesting shapes. The human face is one of the most popular subjects in art. I have created the following drawings by using mathematical formulas. Their formulas are a bit long. So I hope to find simpler formulas in the future. However, I love them because I believe they are very beautiful.

#### Face (1)

This image shows 12,000 circles. For k=1,2,3,...,12000 the center of the k-th circle is (X(k), Y(k)) and the radius of the k-th circle is R(k), where

X(k)=(-4/5)sin(πk/6000)+(1/10)sin(πk/150)(sin(πk/12000))^{10}(sin(πk/4000))^{10}(cos(πk/1000))^{10},

Y(k)=-cos(πk/6000)-(5/6)(cos((πk/16000)-(3π/8)))^{16}-(4/5)(cos((πk/7500)-(4π/5)))^{20}(sin(7πk/12000))^{20}-(1/4)(sin((πk/3750)-(8π/5)))^{2}sin(πk/5)((1/3)-cos(πk/6000))^{2},

R(k)=(1/50)(cos(2πk/5))^{4}(cos((πk/7200)-(5π/6)))^{2}+(1/15)((1/2)-(1/(2sin(2π/5)))sin(πk/5))^{20}((sin(πk/12000))^{10}-(sin(πk/12000))^{30})+(1/5)(sin((πk/10)+(10π/31)))^{10}(sin((πk/5)+(10π/31)))^{10}(sin((23πk/60000)-(23π/10)))^{20}(sin(πk/12000))^{4}.

#### Face (2)

This image shows 12,000 circles. For k=1,2,3,...,12000 the center of the k-th circle is (X(k), Y(k)) and the radius of the k-th circle is R(k), where

X(k)=(-4/5)sin(πk/6000)+(1/5)sin(πk/150)(sin(πk/12000))^{10}(sin(πk/4000))^{10}(cos(πk/1000))^{10},

Y(k)=-cos(πk/6000)-(5/6)(cos((πk/16000)-(3π/8)))^{16}-(4/5)(1+(1/15)cos(πk/25)(cos(3πk/2000))^{12})(cos((πk/7500)-(4π/5)))^{20}(sin(7πk/12000))^{20}-(1/4)(sin((πk/3750)-(8π/5)))^{2}sin(πk/5)((1/3)-cos(πk/6000))^{2},

R(k)=(1/50)(cos(2πk/5))^{4}(cos((πk/7200)-(5π/6)))^{2}+(1/15)((1/2)-(1/(2sin(2π/5)))sin(πk/5))^{20}((sin(πk/12000))^{10}-(sin(πk/12000))^{30})+(1/5)(sin((πk/10)+(10π/31)))^{10}(sin((πk/5)+(10π/31)))^{10}(sin((23πk/60000)-(23π/10)))^{20}(sin(πk/12000))^{4}.

Also, in my previous posts you can see more images that I created with mathematical formulas: *Drawing Plants With Mathematics*, *Drawing Birds in Flight With Mathematics*, *Using Mathematical Formulas to Draw Butterflies*.