Lindy Hoppin' Linus

I found the number phi in Boston Common the other day while wandering the city of Boston on a lovely Friday afternoon and strategically trying to avoiding the Black Friday crowds. [Not quite sure it worked...but that's an entirely different topic.]

I was walking past the Visitors' Center on the Tremont St side of the park, and happened to glance up at the sculptures that border the little circle in front of the Visitors' Center building. I was face-to-face with the one entitled Industry. And what do you suppose the man in the sculpture was working on? A dodecahedron!!! [Not sure what a dodecahedron looks like? Click here for a good picture and description.]

Who would have thought I would find such a fabulous polyhedron just hanging out in a sculpture in the middle of the city of Boston?

I suppose, however, that I should give a brief explanation of phi and it's link to the dodecahedron. I recently finished reading The Golden Ratio: the Story of Phi, the World's Most Astonishing Number, by Mario Livio, so the explanation is quite fresh in my mind. [And it's actually quite a simple relationship, without a lot of bells and whistles. The amazing properties of the number phi make the number infinitely cooler than it may sound here. Look it up, it's fabulous!]

The diagonals of a regular pentagon (all sides of the same length, all angles equal to 108 degrees) cut each other in what Euclid defined as an "extreme and mean ratio" (which later became known as the Golden Ratio or the number phi, equal to 1.61803399...). Thus, you would use phi in the geometric construction of a regular pentagon. The dodecahedron is a twelve-sided object whose twelve faces are all regular pentagons. The number phi is literally bursting out of the dodecahedron from every side!

And I suddenly feel quite a different connection with the Common because of this find. The Common is suddenly connected much more to my world of mathematical images and patterns and structures. I love seeing geometrical structures in architecture as I wander the streets of Boston and Cambridge. I pick out patterns in just about anything that can be formed into a pattern (including random patterns in asphalt, brick, cobblestones, etc.). And I love bumping into shapes and 3D geometrical objects in unexpected places when it's obvious someone explicitly put them there (in other words, not by natural causes - there was a plan for the placement of said shape or object).

I find it fascinating that a familiar place can change so drastically depending on the perspective with which I'm looking at it. Day vs. night, time of year, other people dwelling there, personal circumstances and moods. All of these affect one's view of a place. And it changes its appearance, feel, scent, sound, mood with these different perspectives. But, with all of that, a place can take on a whole new meaning to me with one little discovery. Truly amazing...

P.S. - Some of my favorite properties of this fabulous number phi are:
- [phi]^2=[phi]+1 (phi squared equals phi plus one)
- 1/[phi]=[phi]-1 (one over phi equals phi minus one)
- the Fibonacci numbers are intricately related to phi
- phi is found in the pattern of placement for a rose's petals, a sunflower's seeds, a fern's leaves, and a nautilus's chambers on it's shell
- other places that phi pops up: the pentagon, the pentagram (a regular star: the diagonals of a pentagon), the icosahedron (pretty much the "opposite" geometrical object of the dodecahedron), Penrose tilings, quasi-crystals...the list can go on and on...

Seriously. Look it up. It's a truly remarkable number!