Geometric significance of the binomial expansion

Parametric equations

$(a+b{)}^2=a^2+2ab+b^2$ is a square of side $a+b$ cut into four blocks. $(a+b{)}^3$ is a cube of side $a+b$ cut into eight blocks and drawn as an oblique projection, not an unfolded net. In the expansion, $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ counts the ways to choose $k$ factors equal to $b$ and leave the rest equal to $a$. The picture shows $n=2$ and $n=3$ only.

Each block is labeled with its monomial, such as $a^2$ or $a^{2}b$. The binomial coefficient is not written on the block. It is how many blocks share that monomial: one $a^2$, two copies of $ab$, and one $b^2$ in the square; one $a^3$, three $a^{2}b$, three $a{b}^2$, and one $b^3$ in the cube.

Interaction

• n = 2 square: show the square cut into $a^2$, two copies of $ab$, and $b^2$
• n = 3 cube: show the cube, in oblique projection, cut into $a^3$, three copies of $a^{2}b$, three copies of $a{b}^2$, and $b^3$
• Side a: change one length, from 1 to 10. The blocks change size. The labels stay the monomials
• Side b: change the other length, from 1 to 10, in the same way

What to notice

• Two blocks labeled $ab$ are the coefficient 2. Three blocks labeled $a^{2}b$, and three labeled $a{b}^2$, are the coefficient 3.
• Those counts are row 2 and row 3 of Pascal’s triangle. This picture does not draw the triangle.
• The area or volume of a block is not printed. Only the monomial is. Changing the lengths changes the sizes, not the words on the blocks.

Related

• Pascal’s triangle
• Path counting
• Catalan numbers

Further reading

• Binomial theorem (Wikipedia)