'Stencils are cut out of thin board with different curves as regular sine curves---/',


Fig.104. Parallel sets of regular sine waves drawn freehand create an undulating pattern which, with attention, seems also to oscillate in the third dimension.

'---/wave curves of decreasing amplitude,---/'


Fig.105. Parallel sets of sine curves of rapidly decreasing amplitude oscillate in two dimensions and with concentration also three-dimensionally where one sees cavities and protuberances.

'---/ wave curves of decreasing frequency,---/'


Fig.106. Sine curves of decreasing frequency oscillate in two dimensions but also with concentration on their inward and outward movements in the third dimension. This even suggests a force operating from the left.


Fig.107. Curves of decreasing frequency which are out of phase, being darkened alternately, create a concrete twisting effect giving a solid volume as well as a pattern.


Fig.108. Curves out of phase show the amplitude decreasing toward the left while the frequency decreases toward the right. The alternate filling-in creates both a twisting effect and complex oscillations outward or inward with, at some points, a stable pattern.

'---/ cycloids, ---/'


Fig.109. Cyclo´ds are curves traced by a point situated on the radius of a circle as this circle rolls along a straight line. These seem to oscillate simply with a definite rhythm.

'---/ trochoids etc (see below).---/'


Fig.110. A trochoid generated by a point in one curve that rolls along another, is here exhibited as rolling within an exterior circle.


Fig .111. A pentagonal trochoid executed with a compass within a series of convergent circles(themselves not shown but only indicated) demonstrates a three-dimensional instability, oscillating forwards and backwards.


Fig.112. When the trochoids are drawn freely by hand, this oscillation is much intensified, the trochoids seeming to emerge in three dimensions out of the base or to go into it. The free drawing hand picks up and emphasizes this geometrical movement.

Patterns of interferences are studied. How they interact when superimposed in series of horizontal curves over series of vertical curves as well as in rotational variations - is studied and found to give surprising results - both coherence and turbulence.
'---/Using these stencils, sheets are covered using a systematic rule as repeated vertical-horizontal---/'


Fig.113. Parallel sets of sine waves- vertical and horizontal- create an oscillating field coherently vibrating.


Fig.114. Parallel sets of sine curves of decreasing amplitude in the vertical and horizontal directions create a paradoxical situation: whereas in the lower right section they oscillate giving a coherent field as they approach the upper left corner, due to their overlapping, this coherence is lost and they become either flat or turbulent. In this diagram one can see the flatness arising out of the lower right's coherent oscillations.


Fig.115. Parallel sets of sine curves of decreasing frequency vertically and horizontally directed present a similar situation: while in the lower left section the sets oscillate moderately giving a coherent field, they become incoherent - creating a two-dimensional flat pattern due to their beginning to overlap in the upper right section.


Fig.116. Sets of horizontal curvilinear waves of decreasing frequency are viewed against such vertical sets. This reads as an oscillating field in the lower right corner and with increasing steepness toward the upper left it looses its undulating field quality and becomes incoherent - a flat pattern without depth.


Fig.117. Although a simple set of repeating cyclo´ds does not yield much that is surprising, superimposing them with vertical repeated cyclo´ds leads to optical perplexity. Their interference leads to bulging which can, with conscious control, be seen to oscillate in and out, the change being accomplished by means of staring either at the horizontal component or at the vertical. This is not a gentle oscillation which can be seen passing between its opposites but rather a pulsing, in one position or in the opposite without passing through from one to the other. It has a similarity with the double cube which can be seen as projecting or going inward by a change in Gestalt but the passing between them is not to be observed.

'---/diagonal-rotational displacement.'
Beyond the simple operation of lining up sets of curves perpendicularly in vertical and horizontal directions, the next most direct act consists in arranging them at equal distance around a centre - a simple operation which leads to surprising consequences. Although there is no actual interference here there is regular circular movement between inward and outward.


Fig.119. Sine curves of decreasing frequency with diagonal and rotational repetition.


Fig.120. Sine curves of decreasing amplitude going toward the centre in a rotational-diagonal repetition creates three dimensional circles around the centre which ascend on the right and descend on the left lending a concrete suggestion of the activity of surfing in waves.


Fig.121. Cyclo´d waves of decreasing amplitude going toward the centre in diagonal/rotational repetition create a bulging movement on the right leading naturally to a concave movement on the left. The inside circle appears to volume out on the left and contract on the right, the whole moving in a clockwise direction.