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FOUR
 Cube
within sphere
A
sphere fits inside a cube. The volumes of this sphere and cube must therefore
be as
 to 6. If the cube is inside
another sphere, see if you can show that its surface area is thrice that
of the sphere within. |
 House
of Pi
A
square fits inside a triangle and a sphere. Dividing the circle into twelve,
we notice how the triangle and square fall on these points. The square
turns out to have
times the area of
the circle. It divides the triangle into a top part which has one-third
the area of the whole. It's a theorem not in Euclid, to be sure (Ground
measurements of the formation were made by Andreas Muller, as confirm the
reconstruction here given). |
| Lino Floor Pattern
8 June 1990 Exton, Hants
Back
in 1990, croppies were evaluating the quintuplets: patterns of four surrounding
a central circle. The Circlemakers were exploring the idea of tangents,
of lines that just touched.
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The square pattern
set up by the four tangents is pretty obvious, but then a more remarkable
octagon - pattern emerges. It looks as if all of the small triangles are
identical in this diagram. For that to be so, the centres of the small
circles have to be at a distance of twice the radius of the big one, from
its centre. The circle-sizes are in the ratio 1 : 1 +
 2.Thus
the first-beginning of Hypermaths appeared as delicately touching tangent-lines,
noticed by John Martineau. I consulted circle-geometer Allan Brown over
this, and he agreed that this early formation would bear the above interpretation
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 This
one's my favourite. It comprises merely a 3:4:5 triangle, rotated through
four steps. Or, a rope is divided into eight equal sections with knots,
each five units as shown on the grid, pegged out to make the pattern. Or,
one starts from the enveloping square around the perimeter which
is divided into eight equal sections, then moving around, three sections
at a time, will weave it out. Why should doing this produce a 3:4:5 triangle?
It does, that's for sure. The eight little triangles around the centre
are six times smaller than the big ones, and the two central squares have
areas one- fifth that of the enveloping square. An asymmetric octagon
forms its centre. Or, is it woven out of rhombi, of four rhombus-shapes? The
picture (the School of Athens, by Raphael) shows Euclid sketching
out what must surely be, the Compass Rose.
|
 Maltese
Cross 4 August 1999 West Kennet Longbarrow,
Wilts
This
iterative pattern had square sizes decreasing as one, ½ and 1/6th,
each with its centre on the corner of the bigger one. This made their total
area double that of the main square (checkout its equation, A + ¾(3A/4
+ 12A/36) = 2A where A is area of the main square) - and, its total perimeter,
just treble! The Circlemakers have a remarkable knack of finding these
whole-number solutions. Persons without interest in mathematics merely
derive an aesthetic delight from the shapes, without apprehending the just-so
nature of the ratios. This design enjoys both rotational and reflective
symmetry. The crop was flattened in two different directions giving the
remarkable 3-D effect www.korncirkler.dk/cccorner/universe3.html.
A sister-formation appeared a month earlier: www.korncirkler.dk/universe/windmill2.jpg
(July 16).
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Hexagon
Brooch 22 August 1999
 Looking
at this formation gave a 3-D effect, as if one were seeing a hexagon in
perspective. 'The formation itself
was quickly regarded as another optical illusion design, which has been
a major theme for the 1999 season,' commented Stuart Dike. At
its centre was a square: if the six triangles shown are all equal, as they
appear to be, then they are all golden, i.e. have base angles of 72°
or one-fifth of a circle - (strictly it's 71.6°, the angle whose tangent
is three). |
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  Squaring
the Circle
Wherwell, 1995 The
diagram shown is an 'ancient construction for squaring a circle, creating
a circle whose circumference approximated the perimeter of a square' (Michael
Schneider, A Beginner's Guide to constructing the Universe 1995). It was
a 'traditional diagram of temple foundation in India' (Michell, Dimensions
of Paradise, diagram also from there). The formation had a different
design, with arcs from corners of a square drawn through adjacent corners.
This gives a square and circle having the same area, while the previous
design gave more of an agreement in terms of their circumferences. In both
cases the agreement was within a few %. These two traditional methods of
squaring the circle, by area versus circumference, are treated in Michell's
Dimensions
of Paradise.
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Squaring
the circle II Heabourne worthy July 1997
  The
same area here appears in the central circle and in the sum of four circles
(their outer rims); it is also that of the latent square, as specified
by the four circle-centres. Does this equality also apply to the circle
perimeters? Check this using the accurate silhouette here (drawn by Peter
Sørensen). |
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 Octagon
24 June 1999 West Overton, Wilts
Moving into three dimensions, an Octahedron is made up of triangles. One just needs to cut out the pattern here laid down, and fold it up! NB the smallest units here are all perfect hexagons (Steve Alexander picture, Freddy Silva diagram). See also www.korncirkler.dk/universe/jays1.jpg |
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 Lattice
Pattern 12 Aug 2001 South Yorkshire This
design begins with arcs from four corners of the circle, of the same radius
as it, which meet in the middle (Figure a). Then, if this radius is eight
units, successive arcs are then drawn of seven, six etc. units down to
3. Of especial interest here is the middle symmetry position where four
equal 'octagonal arcs' meet (Figure b).
 These
octagonal arcs will not precisely coincide with the arcs drawn by the above
method, as neither will the small arcs in Figure 'a' (which make one-sixth
divisions of the circle), however the difference is small and within the
tolerance of the thickness of the lines. This construction is thus a study
in approximation. |
 One
can imagine a craftsman and mathematician conversing. The former would
like to make a stained-glass window, and appreciates how the pattern's
strong, fourfold structure will tend to give people a reassuring sense
of stability. She or he would like to divide the radius by eighths for
this construction, and is that good enough? The arcs look as if they are
equidistant, after all. The mathematician appreciates how the sense of
'sacredness' of sacred geometry comes from the way these approximate solutions
are closely converging. He explains how the circle is divided by 1/6th
and 1/3rd parts in 'a' and by octagonal arcs in 'b,' and how
the latter division by 8ths will give the central arcs too small
by a mere 2%, and the smallest arcs (Figure a) to small by 3%, these being
well within the tolerance of the line-width. |
| The
Octagon 3rd July 2005, Alton Barnes
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| Cosmic
Cubes 1st August, 2007 Upper Upham, near
Aldbourne, Wiltshire
The Moon goes round the zodiac every month, and meets up again with the Sun, and each time it does this it has moved round about 30° from its previous position. This might be a helpful analogy. Use of this odd number means that every point gets touched once in 36 steps, i.e., the pattern starts to repeat after 36 moves. Other prime numbers will also do this, whereas even number intervals only touch half of the points, as one follows the line round.
This procedure makes every angle at the periphery equal to 50°. Moving towards the centre, the apex angle of each triangle increases by 10° at each step – so the next one in, had there been another layer, would be 90°: i.e., those straight lines extended, staying within that inner circle, make squares! All the lines meet at right-angles on the innermost circle. A ring of 36 right-angles are surrounding the centre. Thanks to Steve Brown for guidance on this one. |
The number Four, the first square number, reconciles the two forms of mathematical growth, being both 2+2 and 2x2, and it represents the human instinct for symmetry and order by dividing the compass onto four points and the year into four seasons. It is at the foundation of civilisation, settlement and rectangular land division. It is the foursquare number of solid earth as opposed to formless heavens.John Michell, Dimensions of Paradise 1988
Thanks to Sørensen for use of 'Hexagon brooch' photo, Freddie Silva for 'Squaring the circle' photo, to Steve Alexander for 'Octagon' photo, and to Nick Nicholson for 'Lattice Pattern'; to John Martineau for use of his 'Lino floor pattern' diagram, and to John Michell for use of circle-squaring diagram.
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