Four Polyhedra with 7-Fold Pyramidal Symmetry

Posted on 2 January 2023 by RobertLovesPi

I made these four polyhedra using Stella 4d, which you can try for free right here.

Seven Times Two Is Fourteen, Two Versions

Fractiles’ Mandala, Based on Angles of Pi/7 Radians

Posted on 14 June 2016 by RobertLovesPi

$fractiles7withblackbackground$

Although this was based on something I constructed using the Fractiles-7 magnetic tiling toy, I did not have enough magnetic pieces to finish this. The idea was, therefore, converted into a (non-Euclidean) construction using Geometer’s Sketchpad, and then refined using MS-Paint. The reason I describe this as a non-Euclidean construction is that an angle of pi/7 radians, such as the acute angles in the red rhombi, cannot be constructed using compass and unmarked straight edge: antiquity’s Euclidean tools. The other angles used are whole-number multiples of pi/7 radians, up to and including 6pi/7 radians for the obtuse angles of the red rhombi.

The yellow rhombi have angles measuring 2pi/7 and 5pi/7 radians, while the blue rhombi’s angles measures 3pi/7 and 4pi/7 radians. None of these angles have degree measures which are whole numbers. It is no coincidence that 7 is not found among the numerous factors of 360. It is, in fact, the smallest whole number for which this is true.

I have a conjecture that this aperiodic radial tiling-pattern could be continued, using these same three rhombi, indefinitely, but this has not yet been tested beyond the point shown.

Seven Fanciful Designs for Polyhedral Throwing Stars with 7-Fold Dihedral Symmetry

Posted on 27 February 2016 by RobertLovesPi

I made these using Stella 4d, a program available at http://www.software3d.com/Stella.php. Any of these images may be enlarged with a click.

Heptamandala

Posted on 3 August 2014 by RobertLovesPi

Seven Moving Lights in the Sky, the Seven Days of the Week, and Other Significant Sets of Seven

Posted on 10 July 2014 by RobertLovesPi

Have you ever wondered why the number seven appears in all the places it does? We have seven days in the week. Churches teach about the seven deadly sins, and “seven heavens” is a common phrase. There are seven wonders of the ancient world, and seven of the modern world. The number seven has appeared in many other socially significant ways, in societies all over the world, for millennia.

It is no coincidence, I think, that the ancients were able to see seven lights in the sky which are either visible in daylight, or move against the background of “fixed” stars at night. They ascribed great significance to what went on in the sky, since they viewed “the heavens” as the realm of the gods in which they believed. The evidence for this lives on today, in the names of the seven days of the week, and numerous other sets of seven, all over the world.

It is possible to see the planet Uranus without a telescope, but it is very dim, and you have to know exactly where to look. No one noticed it until after the invention of the telescope. If Uranus were brighter, and had been seen in numerous ancient societies, I have no doubt that we would have eight days in the week, etc., rather than seven.