Paul L. RosinDepartment of Computer Science Cardiff University UK
A circular rosette is formed by taking copies of a circle
and rotating them about a point—the rosette's centre ( To simplify matters we shall only consider the first case in which the circles meet at the rosette centre. The points of intersection between the circles is given by where
produces patterns very similar to circular rosettes as shown
in One can speculate that Albrecht Dürer may also have noticed
the similarity. In addition to his artistic work, he realised
that mathematics could provide a powerful tool for the artist,
and was interested in the connections between art and mathematics.
This lead to his becoming an important Renaissance mathematician
(at least in terms of early dissemination of geometry rather
than an extension of the field). In his book The hypotrochoid is not the only curve with similar appearance
to the rosette. In fact, in 1728 Guido Grande published
If the circular rosette is uniformly stretched in any direction
this results in a rosette with an overall elliptical form ( Another variation on the theme of ellipses is given in the pavement in the Campidoglio which was designed by Michaelangelo (although not executed until 1940). The final version shows a more subtle construction than the above uniform stretching. The inner ring is a circle and successive rings are increasingly stretched so as to provide a gradual transition to the outer ellipse. Other patterns can be constructed in a similar way to the
circular rosette, but modifying the positions of the circles
and/or their size. For instance, the nephroid and cardioid in
Even simplifying the pattern to a single rosette with infinitely thin boundaries, and butting the ellipses right up to the circles still results in a difficult geometric pattern to analyse. The principal problem involves the ellipses since determining inscribed ellipses is not straightforward. To simplify analysis we approximate the curvilinear rhombus
by a standard straight-sided rhombus ( As A couple of simple corrections were tested to see if there
was an easy procedure for inscribing the ellipses more accurately.
The first correction is based on two corners of the rhombus that
are equidistant from the rosette centre. Our original method
effectively takes their average as the centre which is therefore
placed closer to the rosette centre than the corners. Since it
was seen that this was actually too far in, we considered pushing
the ellipses out so that their centres become equidistant with
the rhombus's corners. This was done by taking one of the corner
points and rotating it to become aligned with the ray through
the centre of the rhombus. A second approach to the correction is to consider the maximum
error between the true circle forming the interstice and the
straight line approximation. This is where
The inscribed ellipses can be considered as being packed into
the lunes, and their centres will lie along the middle of the
lune—its so-called medial axis. As a simpler problem we
consider lunes with inscribed circles as shown in Of course, this degree of separation does
not occur for a rosette since the most extreme case occurs when
there are only three circles (although there is then no rhombus-like
interstice to inscribe the ellipse within). The separation between
adjacent circle centres is then 3
Perhaps the most famous instance of inscribed circles is that
of Pappus of Alexandria, who over two thousand years ago described
how to inscribe circles into the where The equations of the circles inscribed in a lune can also
be determined, although the process is more laborious (see the
Appendix). This enables us to insert circles in the lunes formed
by the intersecting circles making up circular rosettes, as shown
in figure 15. The rosettes contain two sets of lunes; More practical applications of inscribed circles frequently
occur in industrial design, often as a means of providing strength
while minimising weight or material. A pair of contrasting examples
can be seen in eighteenth century bridge design. In the iron
bridge at Sunderland shown in
and its effect is to transform (diagonal) straight lines to equiangular spirals. This enables us to map translational symmetries into rotational symmetries. Figure 17a shows seven columns of circles which are mapped
to the seven concentric rings of ovals in figure 17b. For display
purposes the top half of the columns have been clipped. It should
be noted that the ovals are egg-shaped rather than elliptical
since they contract as they approach the centre of the figure.
The circles in each column lie in the range [0, 2p];
increasing the number of circles increases the radial resolution.
Columns of circles can be added on both sides of
thereafter we can determine the remainder of the sequence of circles as where
return to
text[2] The conjugate diameters [3] The very first iron bridge ever built (by Abraham
Darby III at Coalbrookdale in 1779) is of fairly similar design
but only contains a single inscribed circle. [4] The story goes that William Edwards the builder
made three attempts to build the bridge. The first bridge was
swept away by a flood shortly after completion. Nearing completion
the pressure of the heavy work at the spandrels caused the second
to spring up in the middle. This lead to the final and successful
result which was the longest single span bridge in the UK for
half a century. Not only is this considered by many people to
be the most beautiful arch bridge in the UK, but it is of significant
engineering interest, and has gathered considerable historical
and scientific analysis [Hughes et al. 1998].
Billings, R.W. 1851. Browning, H.C. 1996. Dixon, R. 1991. Dörri, H. 1965. 100 Great Problems of
Elementary Mathematics. New York: Dover. Dürer, Albrect. 1977. Emerson, Ralph Waldo. 1920. Circles. In Gardner, M. 1965. The superellipse: a curve
that lies between the ellipse and the rectangle. Goodyear, W.H. 1891. Heilbron, J.L. 1998. Geometry Civilized: History, Culture, and Technique. Oxford: Oxford University Press. Hughes, T.G., G.N. Pande and C. Sicilia. 1998.
William Edwards Bridge, Pontypridd. Kappraff, Jay. 1999. The Hidden Pavements
of Michelangelo's Lurentian Library. Kober, H. 1957. Melnick, M. 1994. Manhole Covers. Cambridge,
MA: MIT Press. Nicholson, Ben and Jay Kappraff. 1998. The
hidden pavement designs of the Laurentian library. Pp. 87-98
in Phillips, G. 1839. Rawles, B. 1997. Schmelzeisen, K. 1992. Weisstein, Eric W. 1998. Williams, Kim. 1997. Williams, Kim. 1999. Spirals and rosettes
in architectural ornament. Nexus Network Journal
Copyright ©2001 Kim Williams top of
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