1984

### Examples of Integrated, Multi-set Concept Schemes (Annex 20)

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See other Examples of Integrated, Multi-set Concept Schemes. The concept scheme described here is discussed in the paper on: Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation. This was prepared for a sub-project meeting of the Forms of Presentation group of the Goals, Processes and Indicators of Development (GPID) project of the United Nations University (UNU). The annexes were published in Patterns of Conceptual Integration. Brussels, UIA, 1984, pp. 161-204

These patterns result from the oscillation of a plane metallic sheet covered with a suitable fine powder. The patterns are determined by the frequency of oscillation and the shape of the plate and the manner of its support. The patterns are of interest because they indicate ways in which a zone can be broken down "naturally" into sub-zones under different conditions. The items below were extracted from:

• Mary D Waller. Chladni Figures; a study in symmetry, London, G Bell, 1961

0.1 "Similarly on plates which have only one line of symmetry, the nodal systems may be placed in one of two classes of vibration pattern symmetry. In one of these classes the symmetry line is nodal while the other class is antinodal. The displacements of the surface at corresponding points on either side of the line of symmetry are equal to one another; when the line is nodal they are out of phase, while when it is anitnodal they are in phase. The displace- ments of the vibrating plate at right angles to the equilibrium plane are thus at every instant 'mechanically balanced'." [p. 14]

0.2 "The normal modes of vibration of all plates (of all shapes) correspond with one another" (p. 26)

0.3 "For a plate with 1 line of symmetry there are 2 classes of vibration pattern symmetry...When 2 lines of symmetry are present there are 4 classes....From further observation on shapes...it may be concluded that a particular vibration yattern of a plate with n lines of symmetry can be classed according as it possesses 1, a, b,....n lines of symmetry where 1, a, b,....n are the factors of n. Since each line of symmetry can be either nodal or antinodal it follows that: The normal nodal systems of a plate which has n lines of symmetry may be divided into classes which are twice as numerous as the number of factors of n." (p. 28)

 n Line symmetry Factors of n Number of factors f Number of classes = 2f 4.1 Triangle (equilat) 3 1,3 2 4 4.2 Pentagon 5 1.5 2 4 6.1 Square 4 1 ,2,4 3 6 8.1 Hexagon 6 1,2,3,6 4 8 6.2 Octagon 8 1,2,4,8 4 8 8.3 Decagon 10 1.2,5,10 4 8 12.1 Dodecagon 12 1.2,3,4,6,12 6 12
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