Valentine’s For That Special Someone

     Nothing says “I love you” like a non-orientable surface. Instead of just making the usual Möbius strips for Valentine’s
     Day though, this year, consider making interlocking Möbius based Hearts!

Möbius Love Bands Möbius Hearts

Instructions for making the topologically fun Hearts can be found from Matt Parker or on the 360 Blog.

Topologically, the Möbius strip can be defined as [0,1] × [0,1] with its top and bottom sides identified under the relation (x, 0) ~ (1 - x, 1) for 0 = x = 1

The Möbius strip is a two-dimensional compact manifold with a single boundary. Cutting the Möbius strip along the center line creates one long strip with two full twists in it, and two edges.

     While the topology of a Möbius strip sliced down the centre line isn’t as interesting (as it’s no
     longer a Möbius strip then), the gesture still seems sound. The Möbius based hearts are a nice
     addition to any Valentine's Day.

     There is also the Sierpinski themed option, with instructions available on the 360 Blog as well.

Sierpinski Popup Card

Based off of the Sierpinski triangle, the pop-up Sierpinski heart card is a romantic homage to the famous fractal.

The basic idea behind the creation of the pop-up heart card is a simple application of the standard algorithm for obtaining arbitrarily close approximations to the Sierpinski triangle. Unfortunately, in the paper-heart case, the difficulties of folding come into play.

When done properly, the number of hearts (triangles) after the nth iteration is 3^n.

While it’s not the most careful application of the fractal generating algorithm, it’s still pretty cute. Although, since the (Lebesgue) area of a Sierpinski triangle is zero, the romantic gesture might be a little lacking.

Loveable Equations

     The Wolfram|Alpha blog did a nice post for this Valentine’s Day featuring some very loveable equations. Instead of paper, try generating some your favourite
     shapes with the equation grapher of your choice.

     There are five standard one dimensional curves (two dimensional graphs) for Valentine's Day: the (rotated) Cardioid, the First Heart Curve, the Second
     Heart Curve, the Third Heart Curve, and the Fourth Heart Curve.

Rotated Cardioid First Heart Curve Second Heart Curve Third Heart Curve Fourth Heart Curve

     Of course, there are always the standard two Valentine’s Day Surfaces to choose from:

Taubin's Heart Taubin's Heart Surface
Or the Bonne Projection Bonne Projection

     If all else fails, candy is still a valid fall back option.

Candy Hearts

Happy Valentine's Day!

© 2010 S.C. Kavassalis