Circumparabolic Regions Inside a Unit Circle

Posted on 22 January 2016 by RobertLovesPi

circumparabolic regions

A circumparabolic region is found between a circle and a parabola, with the circle being chosen to include the vertex and x-intercepts of the parabola used, with the circle, to define the two circumparabolic regions for a given parabola-circle pair. There are four such regions shown above, rather than only two, because two parabolas are used above. The formulae for the parabolas, as well as the circle, are shown.

A puzzle which I will not be solving, I suspect, until I learn more integral calculus: what fraction of the circle’s area is shown in yellow?

87 Rotating Non-Convex, Non-Chiral Polyhedral Images Featuring Icosidodecahedral Symmetry, Plus Four Which Snuck In with Cuboctahedral Symmetry — Can You Find All the Intruders?

Posted on 22 January 2016 by RobertLovesPi

To see larger versions of any of these, simply click on the images.

Excavated GRID with triangular cupola and more embellishments

Dodecahedra 5 compound with icosahedra 5

compound great icosahedron and dodecahedron

Compound of Trunc Icosidodeca and Rhombic Triaconta

207th Stellateion of the Rhombicosidodeca converted to base model and colored by face type

Faceted Dodeca and should be a uniform polyhedron

Faceted Icosidodeca six reg decagons going through center twelve star pentagons 30 golden rectangles nth stellation more n and more and more

Faceted Icosidodeca six reg decagons going through center twelve star pentagons 30 golden rectangles nth stellation more n and more

Faceted Icosidodeca six reg decagons going through center twelve star pentagons 30 golden rectangles nth stellation more n

Faceted Icosidodeca golden rectangles dodecagons and star pentagons

24 to this point….

Faceted Stefdhllated Rhgdhombic Triaconta

Faceted Stellahhctejhfgfd Facjhfjhfeted Dual

Faceted Icosidodeca six reg decagons going through center twelve star pentagons 30 golden rectangles nth stellation

Faceted Stefdhllated Rhgdhombic Triaconta base dual compound

Faceted Triakisicosa which is the dual of the 22nd stellation of the truncated dodecahedron

Faceted truncated icosahedron the dual made of sixty intersecting and irregalar hexagons

Facetedfdjsjh Shgdytelhgytlated Faceted Dual

That’s 40 so far…

$fractured great dodecahedron$

great rhombicosidodecahedral shell produced by drilling

Now the count is at four dozen.

That was 26 more, so there are 48 + 26 = 74 so far.

stellated icosa final stell dual a faceting of the RTC rcm

stellated icosa final stell dual a faceting of the RTC

Stellated Icosidodeca fs colored by i dunno

Faceted Rhombic Triaconta with Four Faces Meeting At an Edge

Faceted Trunc Icosidodeca COMPOUND 15 CUBOIDS FIVE-COLOR VERSION DUAL

Faceted Truncated Icosidodecahedron / Compound of Fifteen Cuboids

Now the count is up to 83.

Faceted Poly GRID compound of six decagonal prisms

Faceted Rhombicosidodeca compound of six elongated pentagonal prisms the dual six trapezohedra

Faceted Rhombicosidodeca compound of six elongated pentagonal prisms

Faceted Trunc Icosidodeca COMPOUND 15 CUBOIDSCOLORED BY FACE TYPE DUAL

Compound of enantiomorphic pair of faceted snub dodecah

34th stellation of the Small Ditri Icosidodeca

So there were 91 of these stored on my hard drive, from all my “hard work” playing with polyhedra using Stella 4d: Polyhedron Navigator. (It will be good for my computer to get all that hard drive space back!) If you’d like to try playing with the same program — for free — just try the free download at http://www.software3d.com/Stella.php.

Three Polyhedra Featuring Heptagons

Posted on 22 January 2016 by RobertLovesPi

Stellated Poly which is chiral and has tetrahedral symmetry and features hexagons and heptagons and squares

chiral polyhedron featuring a dozen heptagons

For some reason I do not fully understand, polyhedra featuring heptagons, even if irregular, do not appear often, at least not in my geometrical investigations — so I was pleased to find these three, using Stella 4d: Polyhedron Navigator, available at http://www.software3d.com/Stella.php.