Theoni Pappas - Mathematical Snippets: Exploring mathematical ideas in small bites

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To the mathematician in all of us.

Its there.

Do the math!

Whats a mathematical snippet? It may be an idea, a problem, a puzzle, a piece of art, a text bitegiving you a glimpse into the vast world of mathematical ideas. The snippet is just a portal of discovery, a mere snapshot of mathematics at work. I barely touched the surface of a mathematical concept, but hopefully captivate you enough to explore it further.

Many famous attractions world wide have mathematical connections. St. Peters Square in Vatican City is one such site. Although thousands of people visit it daily, only a small fraction of its visitors zero in on the mathematics of this famous piazza.

Its designer, Gian Lorenzo Bernini (1598-1680), used properties of a trapezoid, an ellipse, a circle and perspective to create a welcoming square which can comfortably accommodate over 300,000 people. Completed in 1667, the oval shaped piazza is embraced by two curved colonnades consisting of four rows of columns. The two colonnades were erected on opposite ends of the ellipses major axis of 650 feet. Bernini used imaginary trapezoidal shapes within the piazza. The front of the Basilica and the ellipses major axis (opposite parallel sides of the blue trapezoid) form one trapezoid. Within it are the green and red trapezoids. By creating these imaginary trapezoids the perspective one would naturally experience when entering a rectangular space becomes exaggerated and enhanced by the trapezoids and additionally directs the visitors attention to the Basilicas facade. Within the elliptically shaped piazza two fountains are located on opposite sides of an obelisk which stands in the piazzas center. The fountains were positioned so that they are located at the focal points of the ellipse. Located between each focus and the obelisk are two disks. Stand at any location in the piazza other than on these two disks, and the front of columns and other rows of columns in colonnade are visible. However, if one stands on either of the disks marked with the phrase centro del colonnaio all the columns behind the front row disappear. Its both startling and intriguing. Illusion or mathematics?

From a satellite view of St. Peters Square. Note the two fountains straddling the obelisk in the center of the piazza. The colored segments indicate the various imaginary trapezoids.

Photo of St. Peters Square taken while standing off to the side of the disk marked centro del colonnaio.

One of the disks between the obelisk and the fountain which is the center for the imaginary four concentric circular arcs of the colonades columns.

Photos of St. Peters Square taken while standing on the disk marked centro del colonnaio.

Whats the mathematics behind this vanishing act? It all has to do with the properties of a circle. Imagine standing at the center of four concentric circles. Shown here are four concentric arcs of columns. Radii emanating from the common center contain all four columns. If a person is at the circles center, the columns on the line of radius are obscured by the front column on its radius. The viewers line of perspective makes the other three columns behind the front one vanish behind that column.

This example illustrates how different voting systems can affect the outcome of an election.

Suppose candidates X, Y and Z have 35%, 32% and 33% of the votes respectively.