Crossbeams as a Differential Network

Charles SharmanMarch 30, 2015


I analyze Crossbeams for its combinatorial degree and compare it with other networks, finding Crossbeams exceeds most human-designed networks and approaches living systems in combinatorial degree.


Changizi et. al. analyzed various networks for their combinatorial abilities [1]. Network complexity roughly follows

E ∼ Cd

where, E is network complexity, C is number of nodes, and d is the combinatorial degree.

They found living networks (organisms, ant colonies, and nervous systems) have combinatorial degrees (d) from 4 to 18, while human-involved networks (electronic circuits, Legos, businesses, and universities) have combinatorial degrees from 1 to 3. Legos have a combinatorial degree of 1.41.

Crossbeams' Combinatorial Degree

According to Changizi et. al., a building system's complexity follows the number of pieces, and a building system's number of nodes follows the number of piece types. Thus, a building system's combinatorial degree can be derived by plotting the number of piece types versus the number of pieces on a log/log plot and fitting a linear regression.

I performed the analysis for Crossbeams over the first 83 kits released.

Crossbeams Complexity

Crossbeams has a combinatorial degree of 6.07 (the inverse of the slope), which greatly exceeds most human-designed networks and approaches living systems. Crossbeams high combinatorial degree comes from the number of expressions in each connection. That is, Crossbeams connect in up to eight orientations. My early Crossbeams brainstorms, which allowed connections in only two orientations, had more than 130 frame piece types. With eight expressions, Crossbeams has 18 frame piece types and builds the same models.

The low R2 may concern some. The high combinatorial degree (a nearly horizontal line) reflects a fairly low correlation between complexity and number of piece types; other piece type factors then cause fit deviations: e.g. moving structures versus stiff structures, and curved shapes versus straight-edged shapes. The same R2 reduction can be seen in Changizi et. al. as combinatorial degree increases.


Crossbeams achieves a high combinatorial degree, exceeding human-designed networks and approaching living systems. In layman terms, Crossbeams can be classified as an excellent building system: one which maximizes possibilities and minimizes component differentiation. It must be cautioned, however, that possibilities does not necessarily correlate to real-world accuracy. A better figure-of-merit probably exists for building toys.


  1. M. A. Changizi, M. A. McDannald, and D. Widders. “Scaling of Differentiation in Networks: Nervous Systems, Organisms, Ant Colonies, Ecosystems, Businesses, Universities, Cities, Electronic Circuits, and Legos” in J. theor. Biol. (2002) 218, 215-237.