Maurits Cornelis Escher (17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically-inspired woodcuts, lithographs, and mezzotints. He was 70 before a retrospective exhibition was held. His work features mathematical objects and operations including impossible objects, explorations of infinity, reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations.

Although Escher believed he had no mathematical ability, he interacted with the mathematicians George Pólya, Roger Penrose, Harold Coxeter and crystallographer Friedrich Haag, and conducted his own research into tessellation.

Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians’ observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real.

Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians’ observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real.

An exceedingly clever visualization of a Cauchy Product.

The first leap of creativity is choosing to represent quantities as black and red squares. Each black square has value 1, each red square has value -1.

The checkerboard is constructed by expanding both series to be multiplied. Each series is written term-wise along the edge of a grid. I.e., one expanded series is written vertically, left of the rows, and the other is written horizontally, above the columns. Each row (or column, cell, etc.) can be obtained by multiplying the corresponding values along the rows and columns.

For example: the upper-left corner cell represents (1)x(1)=1, a black square. The cell to its right (in the same row) represents (1)x(-1)=-1, a red square. The pattern alternates accordingly.

Observe how all the squares along the diagonal, from the corner, are also black. This is because they represent products of matching signs (both positive or negative), and are thus always positive. For similar reasons, all diagonals (and anti-diagonals) have the same color.

The checkerboard represents the left-hand side of the Cauchy product–it is obtained by multiplying two series. The right hand side of the Cauchy product, a single, new series, is represented by the pyramidal structure pointed to by the arrow.

The pyramid is made by reorganizing the checkerboard. To see how, imagine isolating each anti-diagonal. Beginning with the smallest anti-diagonals and working outward, rotate the square(s) and add them together, so they fit side by side (making rectangles). Layer the rectangles top to bottom from smallest to largest.

Since successive diagonals alternate in sign (positive/negative), we obtain a pyramid of alternating black and red layers. Each layer has one more square than the previous. Recalling that black squares are positive 1′s and reds are negative, we can convert the layers of the pyramid into individual terms of an alternating series

which is the right-hand side of the Cauchy product. Incredible!

An exceedingly clever visualization of a Cauchy Product.

The first leap of creativity is choosing to represent quantities as black and red squares. Each black square has value 1, each red square has value -1.

The checkerboard is constructed by expanding both series to be multiplied. Each series is written term-wise along the edge of a grid. I.e., one expanded series is written vertically, left of the rows, and the other is written horizontally, above the columns. Each row (or column, cell, etc.) can be obtained by multiplying the corresponding values along the rows and columns.

For example: the upper-left corner cell represents (1)x(1)=1, a black square. The cell to its right (in the same row) represents (1)x(-1)=-1, a red square. The pattern alternates accordingly.

Observe how all the squares along the diagonal, from the corner, are also black. This is because they represent products of matching signs (both positive or negative), and are thus always positive. For similar reasons, all diagonals (and anti-diagonals) have the same color.

The checkerboard represents the left-hand side of the Cauchy product–it is obtained by multiplying two series. The right hand side of the Cauchy product, a single, new series, is represented by the pyramidal structure pointed to by the arrow.

The pyramid is made by reorganizing the checkerboard. To see how, imagine isolating each anti-diagonal. Beginning with the smallest anti-diagonals and working outward, rotate the square(s) and add them together, so they fit side by side (making rectangles). Layer the rectangles top to bottom from smallest to largest.

Since successive diagonals alternate in sign (positive/negative), we obtain a pyramid of alternating black and red layers. Each layer has one more square than the previous. Recalling that black squares are positive 1′s and reds are negative, we can convert the layers of the pyramid into individual terms of an alternating series

which is the right-hand side of the Cauchy product. Incredible!