Meissner Tetrahedra

Meissner Tetrahedra 3d printed Art Mathematical Art Printed in WSF, then dyed with fabric dye.
Printed in WSF, then dyed with fabric dye.
Meissner Tetrahedra 3d printed Art Mathematical Art Printed in WSF, then dyed with fabric dye.
Meissner Tetrahedra 3d printed Art Mathematical Art Printed in WSF, then dyed with fabric dye.
Printed in WSF, then dyed with fabric dye.
Meissner Tetrahedra 3d printed Art Mathematical Art Showing openings in bottom
Meissner Tetrahedra 3d printed Art Mathematical Art Showing openings in bottom
Showing openings in bottom
Video
3d Model Viewer
Meissner Tetrahedra 3d printed Art Mathematical Art
Meissner Tetrahedra 3d printed Art Mathematical Art
Check out gibell's Reuleaux solids at http://www.shapeways.com/model/115463/reuleaux_solids_3cm.html Those are surfaces of rotation based on Reuleaux polygons, which are curves of constant width. These are also surfaces of constant width, but *not* surfaces of rotation. So despite looking decidedly non-spherical, they roll smoothly as spheres.

There are two different Meissner solids, subtly different.  Look closely at the edges.  Some are rounded and some are sharp.  One of them has the round edges making a a triangle and the sharp edges meeting at a point; the other has them the other way around.

These are sized for compatibility with gibell's solids, so if you get all five they will all roll smoothly under the same flat surface together.  And like gibell's, they are hollow with internal braces to lend extra support, with holes in the bottom to let the support material out.

Printed great.  Video below! (the two foreground shapes are mine; the red one in the background is one of gibell's from the set linked above)

cm: 3 w x 2.998 d x 6.4 h
in: 1.181 w x 1.18 d x 2.52 h

Comments

 
@clsn Thanks a lot, I had found some links about this but most of the time they're about a sentence long without much explanation so thanks for clarifying it for me. (And of course it's only with 3 of the sides; either subtending to a point or the 3 sides of one triangular plane on the tetrahedron). Thanks again.
December 28, 2013, 2:53 pm
@G_GR3G_G , @ekul1018 Take a look at http://www.swisseduc.ch/mathematik/geometrie/bodiesofconstantwidth/docs/meissner_en.pdf for an explanation of how the rounding-off happens. Basically, you extend the planes of the base tetrahedron, remove the solid between the extensions, and complete it again with a rotation of the lines of intersection.
December 18, 2013, 2:25 am
@G_GR3G_G I am also very curious about this.
December 17, 2013, 9:27 pm
the Meissner Tetrahedra (in white), with one of the Reuleaux shapes in blue in the background.
December 8, 2013, 7:37 am
Hi, how did you modify the reuleaux tetrahedron to make this meissner tetrahedron? Obviously you have curved the 3 edges, but what is the radius of those curved edges? is it just curved so that it's smooth to the sides, or is the radius of the curved sections the same radius as the sides of the tetrahedron (reuleaux or meissner)?
December 3, 2013, 7:03 am
@fenixreal The price is for two: one of each of the two slightly different shapes.
December 1, 2013, 3:36 am
The price for the one, or for two? Thank You.
November 29, 2013, 9:33 pm
 
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