# Drawing Human Faces With Mathematics

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)))2sin(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)))2sin(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.