Crystallography & Symmetry
patterns derived from their groups
Concept

Maxwell count — where it appears

A framework's degrees of freedom minus its bar constraints: twice the joints, less the bars, less the rigid motions. It is the number of mechanisms minus the number of states of self-stress, so a framework with one of each reports nothing, and a count of zero settles neither question.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

11 frameworks, 3 where the count is wrong. Every net in this collection read as a framework of rigid bars and free joints, with the cell free to change shape. Maxwell's count and the number of mechanisms agree on most of them and not on all: a framework with a state of self-stress has a bar the count treats as removing a freedom that the others had already removed, and it has a mechanism the count cannot see. Here that is fes, snb, ring5, where the count says 2, -1, -4 and the rank says 3, 0, 9. The identity Maxwell is always right about — count equals mechanisms minus self-stresses — holds on every row.

The count that promises a mechanism

Count the joints, count the bars, subtract. The number that comes out promises rigidity when it is small and a mechanism when it is large, and it is wrong in both directions — because it assumes every bar removes a freedom the others have not already removed.

applied · Rigidity One framework has a count of zero, one mechanism and one self-stress. Every net this collection has a placement for, as a periodic bar-and-joint framework in a fixed cell: its point group, the joints and bars of one cell, the scalar Maxwell count 2n − e − 2, and the mechanisms and self-stresses found exactly from the rank of the rigidity matrix. The scalar count is always the difference of the last two, which is Maxwell's identity — and the bathroom net is the row that shows what the identity costs: nought equals one minus one, and a framework that reads isostatic moves.

The mechanisms a count cannot see

Maxwell's count subtracts constraints from freedoms, and a mechanism and a state of self-stress cancel in the subtraction — so a framework with one of each reports the same number as a rigid one. The bathroom net reports nought and moves. Doing the same subtraction with representations instead of numbers separates them, because a mechanism and a self-stress cancel only when they belong to the same representation.

applied · Rigidity Nine frameworks, counted and then decided. Maxwell's count subtracts bars from twice the joints; the pebble game inserts the bars one at a time and discards any that cannot be paid for. The two agree on most of these frameworks and not on all, and where they differ the count is the one that is wrong — it assumes every bar is an independent constraint, and a bar added to a part that is already rigid is not. The redundant column is how many bars the game refused.

A game that decides what counting only bounds

Maxwell's count subtracts bars from twice the joints and is a bound, not an answer, because it assumes every bar constrains something new. In the plane there is an exact repair: Laman's condition, run as a game in which each joint holds two pebbles and a bar is admitted only if four can be gathered at its ends. Two rigid bodies sharing a joint are what the count gets backwards.

applied · Rigidity

Named alongside it

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

MechanismDegrees of freedomRigiditySelf-stressCharacterCrystal netGram matrixIrreducible representationIsostaticPeriodic frameworkPermutationPoint group

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