the theme
the theme consists of the continuous
development of an equilateral triangle to a regular octagon. in
other words the third side of the triangle which would close it is
moved outward so as to form one of the sides of the quadrilateral
(square). in this way the area of the triangle remains open and is
merely suggested. the transitions from one polygon to the next are all
made in the same fashion. the resultant figure is a spiral composed of
straight lines of equal length. the angles and the areas between these
lines show a great variety of forms and tension.
variation 1
all the regular polygons of the theme are here entirely closed. that
is to say all the lines limiting the areas are put in. one side of each
polygon coincides with a side of the polygon coming after it. the areas
themselves are clearly visible. they are picked out with colours chosen
to correspond with the fundamental forms of the polygons. these colours
are constant and appear in the same order in all the following
variations. although they are clearly recognisable the polygons do not
in reality exist. their places are taken by surfaces of different form
and colour which result from the fact that the area of each polygon has
superimposed on it the polygon which immediately precedes it.
variation 2
the theme is shown in heavy line. it is elaborated by circles drawn in
a fine line. the original lines of the theme form the diameters of
these circles. the interplay of these equal circles is interrupted by
six semi-circles which are drawn heavier. the semi-circles break the
development from triangle to octagon and mark the lines that would have
closed the polygons in the theme by taking these as their diameter.
variation 3
the points in colour mark the angles of the theme. the points in two
colours mark the lines which are common to two polygons and therefore
indicate a change of direction. the colours are developed in the same
order as in variation 1.
variation 4
the angles of each polygon are connected to its center. two of the
tips of each of the resulting coloured stars tuch the tips of the
preceding star. for example the yellow connects with the green and
the green with the turquoise.
variation 5
the circumscribed circles of the polygons are connected to those lines
of the polygons omitted in the theme. the polygons form surfaces whose
smaller sides develop in the same rythm as the theme.
variation 6
the theme is taken up again by the semi-circles which have as their
bases the straight lines of the theme. these semi-circles are described
in alternating opposition to each other. the result is a continuous
movement of changing colours progressing from orange to crimson.
variation 7
the inscribed circles of the polygons are drawn so as to form two
semi-spirals turning in opposition to one another. the grey spiral,
which is moving in the opposite direction to the black, is strengthened
and reinforced by by the angles of the polygons themselves. this
combination gives all the inscribed circles of the complete series of
polygons from the triangle to the octagon.
variation 8
the circumscribed circles of the polygons form segments of circles.
these are shown according to their corresponding colours. those parts
of the circles which are the same length and in which the colours would
be superimposed on each other are left white. the order of the colours
is as in variation 1.
variation 9
a framework is formed by the inscribed and circumscribed circles.
those points where the the cutting of the circles coincides with the
angles of the theme are shown by large spots. the other points where
the circles intersect are marked with smaller spots. the colours are
concentrated at the center of the circles.
variation 10
the inscribed and circumscribed circles produce thick circular rings.
the parts of these rings where no overlap takes place are shown grey.
variation 11
the angles of the polygons are joined to their centers (as in
variation 4). the spearhead shapes thus formed lead from the middle of
the figure to the angles of the polygons and are all shown in a
contrast of black and white. the corresponding colours of the the
polygons appear on areas in which the centres of the respective
polygons occur.
variation 12
the inscribed and circumscribed circles form thick circular rings.
these are shown in their appropriate colours, while those parts where
overlapping occurs are black.
variation 13
the inscribed circles of the polygons touch each other at one point.
the resultant surfaces are shown alternately grey and white.
variation 14
here only the edges of the surfaces formed in variation 1 are coloured.
the result is a great variety of colour arising from the relative
positions of the colours. the star shape of variation 4 are added in
black.
variation 15
the inscribed circles produce a semi-spiral movement formed by the
connecting up of the thick circular rings. the thickness of this
movement is determined by the distance between the inscribed and the
circumscribed circles of the octagon. the various segments of this
semi-spiral form are connected to their center by lines.
max bill
zurich november 1938
back
1935-1938
Fifteen Variations on a single Theme
16 lithographs 12" x 12 5/8".
(30.5 x 32 cm)
Paris: Editions des
Chroniques du Jour, 1938
