Some may find this unique table helpful in understanding spheres.

Beyond the purely mathematic, this fully symmetrical regime may also find application in physics, sociology, ontology and economics.

Note: Using the diametric mode for calculations (mentally or on paper) can be quicker than employing the formal (radial) convention, especially when working with hyper-dimensional domains, exponential growth scenarios, or when one is in need of an easier way to factor between domains of differing dimensionality.

If you would like a personal, complimentary copy of this chart in PDF format (8.5″x11″ – but infinitely scalable) use the form below. Comments optional.
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Notes:

All versions since 2004 have reflected the much needed repair of the broken symmetry found in the 0-sphere definition under the prevailing n-sphere generalisation.

If you prefer the formal mode for transforms between exterior and interior space, simply use •r/d instead of •D/2d.

If you’re a student, check with your professor before applying these principles in your work. If you are the professor, just use your best judgement …and maybe get a second faculty opinion. 😉

Jan. 19, 2011 – Image updated from 2004 version to new 2011 version.

Jan. 22, 2011 – Minor aesthetic changes; image updated.

Jan. 23, 2011 – Diametric ext values adjusted by -1; image updated.

Oct. 26, 2011 – Minor text/aesthetic changes; explicatory notes added.

Dec. 28, 2011 – Declared dimensions as a single-character variable (d);
image not updated — use request form below for most current version.

Jan. 10, 2012 – Image updated.

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