# Dancing Rectangles Model Auxetic Behavior

The Dancing Squares Tessellation can be extended to a tessellation of dancing rectangles. Rectangles may have different dimensions and not necessarily be similar.

### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

 What if applet does not run?

Curiously, such tessellations appeared to be the subject of a very recent material research. Imagine a rubber band that grows fatter when stretched and thinner when released. Such materials, called auxetics, actually exist, but scientists haven't totally figured out how they work. ... German physicist Woldemar Voigt first discovered auxetics in iron pyrite crystals (also known as Fool's Gold) nearly a century ago. His research suggested that the crystals somehow grew thicker when stretched. Voigt could not explain the strange behavior, and no practical applications existed at the time, so researchers ignored the work for decades.

Now, it was observed by the scientists at the university of Malta that the Dancing Rectangles tessellation presents an all-purpose mathematical model of auxetic behavior. The model reveals that when the material is stretched, the rectangles-called rigid, rotating subunits-rotate relative to one another, lowering the material's density but increasing its thickness.

Is not that wonderful?

(My sincerest gratitude goes to John Sharp for bringing this topic to my attention.)

### Plane Tessellations

• Dancing Squares or a Hinged Plane Tessellation
• A Hinged Realization of a Plane Tessellation
• A Semi-regular Tessellation on Hinges A
• A Semi-regular Tessellation on Hinges B
• A Semi-regular Tessellation on Hinges C
• Escher's Theorem
• Napoleon Theorem by Plane Tessellation
• Parallelogram Law: A Tessellation
• Simple Quadrilaterals Tessellate the Plane
• Pythagorean Theorem By Plane Tessellation
• Pythagorean Theorem a la Friedrichs
• Pythagorean Theorem By Hexagonal Tessellation
• Hinged Greek Cross Tessellation
• Pythagorean Theorem: A Variant of Proof by Tessellation