Machine Kinematics
solved, then drawn
Concept

Spanning tree — where it appears

A set of edges of a graph that reaches every node and contains no cycle, so it has one edge fewer than it has nodes. Adding any further edge makes a cycle and removing any edge disconnects it, which is why a spanning tree is the smallest set of connections that leaves nothing isolated.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The mechanism, and the graph that decides how many loops it has. A lazy tong of 5 scissor units drawn over its own joint graph: a node for every body — 10 of them — and an edge for every pin, 13 of those. The number of independent loops is e − v + 1 = 13 − 10 + 1 = 4, which is how many closure equations somebody writing this mechanism out by hand would have to find and is the one quantity in the field that can be read straight off a drawing. It is also all the count knows: Grübler's 4 is 3(n − 1) − 2j and contains no geometry at all, which is why it is right here and wrong four rows further down the ledger. positioned by solving, not by drawing.

The loops are in the graph

Before a network is a mechanism it is a graph, and the one quantity that can be read straight off a drawing is how many independent loops it has: edges less nodes plus one. Grübler's count is that arithmetic and nothing else — which is why it is right about a tong at every size and says a deployable ring cannot open.

networks · Network 5 braces, and it is rigid. A 3×3 grid of squares with 5 of its cells braced by a diagonal, and no freedom left. The bipartite graph on the 3 columns and 3 rows, with one edge per braced cell, has 1 component — and the number of freedoms is one less than that, at every bracing there is. Nothing in the rank computation knows about columns, rows or graphs.

Which diagonal rigidifies a grid

A three-by-three grid of squares needs five diagonals and eighty-one of the hundred and twenty-six ways of placing five will do. Which ones is not a rank question at all: it is whether a graph on the grid's columns and rows is connected, and eighty-one is the number of that graph's spanning trees.

networks · Network

Named alongside it

The objects these essays reach for when they reach for this one.

Grübler's criterionJoint graphMobilityNetworkCircuit rankConstraint jacobianCrease patternDesign ruleEnumerationLoop closureRankRedundant constraint

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