Cut the buckyball in half, move the two halves apart, and add a thin nanotube between them:

It is hard to make a rigid structure with long nanotubes. (The most useful thing you can do with them is bundle them into ropes.) So instead let's use short ones and join them together somehow.

One way to join nanotubes together is to use a saddle shaped section:

You can imagine using this as a strut in some sort of framework. Shorter nanotubes (like the one above) would be fairly rigid, and longer ones would be a bit wobbly. (Really long ones behave like cooked spaghetti.)

For the technically minded, the molecule designed on this site is based on metallic single-walled nanotubes of the zigzag type, with two thirds the circumference of a standard C60 buckyball i.e. (n,m) = (6,0).
(See Equilibrium Structure here for an explanation of the various types of nanotubes).

Join four short thin nanotubes together, using six saddles, into the shape of a caltrop (or a tetrahedron):

Notice how most of the carbon atoms join into groups of 6 (in the shape of a hexagon), but two on either side of the center are in groups of 7 (in the shape of a heptagon). This warps the normally flat graphene sheet and produces the saddle shape. This is just one of many possible saddle shapes, each of which warp by differing amounts. We need this specific one for the next step to work perfectly.

side view

top view

Let's call this a nanojack. (nanocaltrop is too long, and these are similar to jacks)

All the connections between nanojacks in this design are joined this way.

Caltrops were small iron balls with projecting spikes used in medieval warfare. The ground over which an enemy was expected to advance was thickly strewn with caltrops with the effect that advancing horses or bare-footed infantry were quickly disabled by the spikes penetrating their feet.

Caltrops were also used by the colonists of New England who placed them around their villages as a precaution against Indian attack.   (The Probert Encyclopaedia: Warfare)

Join ten nanojacks together into (very) roughly the shape of a sphere:

Notice how this is the intersection of four near perfect circles of nanotubes (actually rounded hexagons, just like beehives). One circle is facing you, and the other three circles are facing away from you. (The centers of the circles form a triangular pyramid, or tetrahedron, just like the arms on a nanojack.)

Join a whole bunch of nanojacks together (26 in fact) and you can see the structure is the same as that of Diamond:

We have formed here a mesh of buckytubes (the other name for nanotubes).

Let’s call it Buckymesh.

You could continue to join nanojacks together like this to make macroscopic objects of close to any shape you wish. Let's call this Basic Buckymesh (on the next page we will design another, much lighter version of Buckymesh).

The molecules most people are familiar with have a few or several atoms in them. Molecules with hundreds of atoms are considered to be large. This molecule can have millions, billions, or trillions of billions of atoms in it. And yet, compared to everyday objects like people, this molecule has very few atoms, so is very light.

Join two nanojacks together - seen from the side it looks like a strange, two-legged dog; seen from the top it looks like a six-pointed star: