The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » Math Topics » geometry.research

Topic: What is Bucky Fuller's 4-D?
Replies: 1   Last Post: Mar 12, 2009 9:23 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Clifford J. Nelson

Posts: 235
Registered: 12/6/04
What is Bucky Fuller's 4-D?
Posted: Mar 12, 2009 8:44 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Quotes from Synergetics to show what Bucky Fuller meant by 4-D.

Fig. 465.01 Four Axes of Vector Equilibrium with Rotating Wheels or
Triangular Cams:

A. The four axes of the vector equilibrium suggesting a four-
dimensional system. In the contraction of the "jitterbug" from vector
equilibrium to the octahedron, the triangles rotate about these axes.

The four axes of the vector equilibrium suggesting a four-dimensional

966.20 Tetrahedron as Fourth-Dimension Model: Since the outset of
humanity's preoccupation exclusively with the XYZ coordinate system,
mathematicians have been accustomed to figuring the area of a triangle
as a product of the base and one-half its perpendicular altitude. And
the volume of the tetrahedron is arrived at by multiplying the area of
the base triangle by one-third of its perpendicular altitude. But the
tetrahedron has four uniquely symmetrical enclosing planes, and its
dimensions may be arrived at by the use of perpendicular heights above
any one of its four possible bases. That's what the fourth-dimension
system is: it is produced by the angular and size data arrived at by
measuring the four perpendicular distances between the tetrahedral
centers of volume and the centers of area of the four faces of the

966.21 As in the calculation of the area of a triangle, its altitude
is taken as that of the triangle's apex above the triangular baseline
(or its extensions); so with the tetrahedron, its altitude is taken as
that of the perpendicular height of the tetrahedron's vertex above the
plane of its base triangle (or that plane's extension outside the
tetrahedron's triangular base). The four obtuse central angles of
convergence of the four perpendiculars to the four triangular midfaces
of the regular tetrahedron pass convergently through the center of
tetrahedral volume at 109° 28'.

Four rotations in the four planes of the four hexagonal cross sections
of the cub-octahedron (VE) will not give gimbal lock.

Give me a fourth gimbal for Christmas meant give me a fourth dimension.

You can divide space into closest packed equal edge length cubes and
label the planes of squares that are perpendicular to three directions
X, Y and Z. There are ninety degree angles between X, Y and Z
directions. The XYZ Cartesian coordinate system uses three-tuples of
numbers (x,y,z), the intersection of three planes, to represent the
location of a point in three-dimensional space.

You can closest pack equal diameter spheres instead of cubes and label
the planes of spheres that are perpendicular to four directions ABCD.
The angle between any two directions A, B, C, and D, is ArcCos[-1/3],
approximately 109 degrees 28 minutes. The Synergetics coordinate
system uses four-tuples of numbers (a,b,c,d), the intersections of
four planes, to represent the location of a point in four-dimensional
space, a regular tetrahedron.

Don't be a square or a blockhead; see:

Cliff Nelson

Dry your tears, there's more fun for your ears,"Forward Into The Past"
2 PM to 5 PM, Sundays,California time,;search_person_id=607

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2017. All Rights Reserved.