SPO Curriculum :: Sphericons

Skills

understanding 3D space
3D modeling with SketchUp
Cutting and cleaning in Netfabb
3D printing

Sphericons

When you rotate a symmetric polygon around a mirror axis, cut the solid into two equal parts and then put the parts back together differently, you produce a sphericon.

Rotating a 5-pointed star about any of its five mirror lines:
Cut the form in half to reveal the original star cross section.

The sphericon is a three-dimensional solid with one side and two edges. It was first introduced by the Israeli game and toy inventor David Haran Hirsch who patented it in Israel in 1980. It was given its name by Colin Roberts, of Hertfordshire, England, who also explored it. It may be constructed from a bicone (a double cone) with an apex of 90 degrees, by splitting the bicone along a plane through both apexes, rotating one of the two halves by 90 degrees, and reattaching the two halves. Generalizing this construction to a regular 2n-sided polygon (square, hexagon, octagon ...), will give access to a set of other 3D solids with the same properties as sphericons when rolling on themselves they cover their whole surface.

Instructions

Creating your regular polygon in SketchUp:

Open SketchUp. Use the millimeter template. Select the figure and delete it
Create a polygon of 5 sides:
$_02_polygon_5.png$
Hold shift so that the polygon remains green and aligned with the origin:
$_03_hold_shift_align_axis.png$
CTRL+click on the figure and Zoom Extents:
$_04_zoom.png$
Select the line tool:
$_05_select_line.png$
Connect the endpoints:
$_06_connect.png$
Continue to connect endpoints:
$_07_connect_these_pts.png$
Delete the excess outer lines:
$_08_select_and_delete_outer.png$
Select the inner hexagon:
$_09_select_inner_hex.png$
Delete the selection. Now draw a vertical line:
$_10_delete_draw_vertical_line.png$
Select half the figure
$_11_select_half.png$
Delete the selection:
$_12_delete_selection.png$
Select the circle tool and change the sides to 64:
$_13_select_circle_change_sides_to_64.png$
Draw a circle on the blue axis at the origin. Then select the face of the circle:
$_14_draw_circle_at_origin_sellect_face.png$
Click on the Follow Me tool and click on the half of the star:
$_15_follow_me.png$

$_16_followed.png$
Delete the circle:
$_17_delete_circle_and_face.png$
Triple click on the form:
$_18_triple_click.png$
Select File>Export STL
$_19_export_millimeters.png$
Name the stl file:
$_20_name.png$

Variations

Here is an eight sided polygon where the line tool connects every third endpoint:
$_01_eight_sides_every_3.png$
Continue the pattern:
$_02_continue_pattern.png$
Continue the pattern:
$_03_continue_pattern.png$
Completed:
$_04_completed.png$
Delete half:
$_05_delete_half.png$
Add squared at each midpoint:
$_06_add_squares_at_midpoints.png$
Continue pattern:
$_07_continue.png$
Delete excess outer lines:
$_08_delete_outer.png$
Delete excess inner lines
$_09_delete_inner.png$
Create a circle on the blue access and select the face
$_10_create_circle_select_face.png$
Click on Follow me and the face of the polygon:
$_11_follow_me.png$
Delete the circle:
$_12_delete_circle.png$

Another variation

Eight sides, every endpoint connects to every other endpoint:
$_every_2.png$

Here is a twelve sided polygon:
$_01_create_polygon_on_axis.png$
Connect every third endpoint
$_01b_create_polygon_on_axis.png$ $_02_connect_every_third.png$
Continue pattern:
$_03_continue.png$
Or connect every fifth:
$_04_same_thing_more_sides.png$ $_05_draw_vertical_line.png$
Select half:
$_06_selec_half.png$
Delete selection:
$_07_delete_half.png$
Delete excess outer and inner lines:
$_delete_outer_then_delete_inner.png$