This section explores one of most amazing aspects of this geometrical tradition: built-in proportional constants that can be constructed in a connect-the-dot format. These kinds of numerical constants are often required learning by students graduating from Middle School. In many Middle Schools the success rates of transmitting these concepts are low; if lucky, they stick in the mind for a week or so, or at least until students have been tested. Many teachers are uneasy about teaching these concepts. The beauty of this elegant geometry is that root proportions and other constants are woven into every piece of geometric art that children are completing. Let's make use of this coexistence of art and proportion.

And so, a challenge: if there is an easier way to introduce these concepts to young minds than the way shown here, let me know. Dinner is on me.

Most of us know about pi, or 3.14..., as the length of the circumference of a circle with a diameter of 1. In the illustrations that accompany this section, the value of 1 is given to the radius instead. Why? Because this is the functional length traditionally used to construct a circle, whether on paper (the compass aperture) or on the ground (the length of rope tied to two poles). This is the length chosen by the circle maker. The functioning radius also determines the length of the diameter. The length of the diameter is a direct ramification of the length of the radius. As such, all numerical models are based on an initial radius of 1. This makes the length of the circumference 6.28... .

Etymologically, diameter can be broken into dia + meter. According to the Oxford Educational Dictionary, the prefix "dia-" sometimes refers to the number two. Translated, dia-meter would mean 2-meter. This would imply a radius of one meter, or unit of measure (be it inch, foot, yard, mile, centimeter, meter, kilometer, cubit, pica, point, etc.)

Epistemologically, the advantage of assigning 1 to the radius is that it provides the young mind with a known concept synonymous with a beginning. The rational and irrational numbers that result from the constructions are thus grounded by a concrete concept already grasped.

Spatially, by the time students are introduced to these often strange numbers, they will have gained familiarity with the regular polygons and their internal structures, the divisions of space that occur within their perimeters. When proportional constants are introduced as specified ratios within these geometrical constructs, they are observed visibly in a natural context - regardless of their irrational or rational quality. A spatial introduction to irrational numbers is far more comprehensive than an introduction of extremely abstract decimal numbers to be remembered by rote. Geometric designs capture irrational numbers in space and expose them to the eye.

Irrational numbers never end, by definition; that is, they cannot be expressed as the ratio of two integers. In the numerical world, ir-rational means "without a ratio." However, in the human world, the term irrational carries an emotional sense that could translate to nuts or crazy - which could describe what happens to some of us when first introduced to these numbers. Geometry certainly holds an advantage over numerical approaches to irrational numbers. A spatial approach to proportional constants and irrational numbers helps to alleviate the traditional nervousness of teachers and students alike. In the spatial context of the circle and its polygons, the functioning of these "irrationals" is elegant, dynamic, and poetic.

The following illustrations and connect-the-dot exercises will help students, teachers, and others to understand the spatial identities and numerical contexts of key proportional constants in an integrated, holistic setting. Discover the scientific magic that unfolds when the regular polygons are addressed numerically in their circular contexts.