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de Villiers
M. (1996)
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The only geometry most people know is the Euclidean
geometry they learnt at school. Furthermore, there appears
to be a belief that the old Greeks and other civilizations
before them had discovered all the geometry there is to
know. Very few realize that many exciting new results in
Euclidean geometry were discovered in the nineteenth and
twentieth centuries, for example, the theorems of Morley,
Miquel, Feuerbach, Steiner, etc.
Table 1 The previous century also saw the axiomatic development of projective geometry whose origins can be traced way back to Pappus (350 AD) and Desargues (1639). A major breakthrough was the discovery and independent proof of the principle of duality by Poncelet, Plucker and Gergonne in 1826. Two theorems or configurations are called dual if the one may be obtained from the other by replacing each concept and operator by its dual concept and operator. In projective geometry we find the following duality: vertices (points) - sides (lines) This duality is strikingly reflected by the projective theorems of Pascal (1623 - 1662) and Brianchon (1785 - 1864) as follows:
Figure 1: Pascal's & Brianchon's theorems Although the initial axiomatic treatment of projective geometry was purely synthetic, gradual incorporation of analytical methods occurred in the latter part of the previous century. Most notably was Klein's famous Erlangen -program (1872) which described geometry as the study of those geometric properties which remain invariant (unchanged) under the various groups of transformations. In short, geometry could be classified according to this view as follows: isometries - transformations of plane figures which preserve all distances and angles (congruency) Since time immemorial, one- and two-dimensional geometric patterns have been used by human beings to adorn their dwellings, clothes and implements. Figure 2a for example shows a Moorish tiling from the Alambra in the south of Spain. The Dutch artist Maurits Escher used tessellations extensively in the production of his artwork in the period 1937-1971 (see Figure 2b for an Escher-like tessellation). Perhaps surprisingly, the study of border patterns and tessellations (tilings) has received unprecedented interest by mathematicians in the twentieth century. Nevertheless, in the seventies a housewife Marjorie Rice discovered four new convex pentagons that tessellate, although mathematicians had thought at that stage that the list of tessellating pentagons was complete (see Schattschneider, 1981). Most recently, Grunbaum & Shepherd (1986) produced a systematic investigation which to some degree equals Euclides' Elements .
Figure 2: Examples of tessellations One of the important concepts in the classification of
border patterns and tessellations is that of
symmetry . Using this concept, border patterns
can be classified into seven different types and
tessellations into seventeen different types. An obvious
property of any tiling is that of a repetition of the
pattern. If a tiling has translational symmetry in two
independent directions it is called periodic .
Although most common tilings are periodic, only about twenty
years ago, the British mathematician Penrose discovered a
surprising set of two quadrilaterals that tile
non-periodically (eg. see Benade, 1995). In fact, it is
still an open question whether or not a single tile exists
with which one can only tile non-periodically.
 
Figure 3: Examples of fractals Recent years have also seen the development and expansion
of Knot Theory and its increased application to biology, the
use of Projective Geometry in the design of virtual reality
programs, the application of Coding Theory to the design of
CD players, an investigation of the geometry involved in
robotics, etc. Even Soap Bubble Geometry is receiving new
attention as illustrated by the special session given to it
at the Burlington MathsFest in 1995. In 1986 Eugene Krause
wrote a delightful little book on Taxicab Geometry,
introducing a new kind of non-Euclidean geometry. Several
international conferences on geometry have been held over
the past decade. In fact, David Henderson from Cornell
University, USA recently told the author that presently they
have more post-graduate students in geometry or geometry
related fields than in pure algebra. 
Figure 4: Soddy center
Figure 5: Gergonne point & Gergonne-Soddy-Incenter line
The Gergonne point G of a triangle is the point of concurrency of the three line segments from the vertices to the points of tangency of the incircle with the opposite sides (see Figure 5a). (The Gergonne point incidentally is just a special degenerate case of Brianchon's theorem). Then as shown in Figure 5b, we find the surprising result that the Gergonne point G, the Soddy center S and the incenter I are collinear. (The outer Soddy center also lies on this line). The author also recently discovered two interesting generalizations of Von Aubel's theorem using dynamic geometry software. This theorem states that if squares are constructed on the sides of any quadrilateral then the line segments connecting the centers of the squares of opposite sides are always equal and perpendicular (see Yaglom, 1962 or Kelly, 1966). After some experimentation, the author managed to further generalize it for similar rectangles and rhombi on the sides as shown below (proofs are given in De Villiers, 1996 & In press). In the Figure 6, EG is always perpendicular to FH . Also KM is congruent to LN where K , L , M and N are the midpoints of the line segments joining adjacent vertices of the similar rectangles as shown. Von Aubel rectangle-generalisation
Figure 6
In Figure 7, EG is always congruent to FH . Also KM is perpendicular to LN where K , L , M and N are the midpoints of the linesegments joining adjacent vertices of the similar rhombi as shown. The "intersection" of these two results therefore provides Von Aubel's theorem. Von Aubel rhombus-generalisation
Figure 7
Just a brief perusal of some recent issues of mathematical journals like the Mathematical Intelligencer , American Mathematical Monthly , The Mathematical Gazette , Mathematics Magazine , Mathematics & Informatics Quarterly, etc. easily testify to the increased activity and interest in traditional Euclidean geometry. The mathematician Crelle once said: "It is indeed wonderful that so simple a figure as the triangle is so inexhaustible in properties ". Perhaps this applies even more widely to Euclidean geometry in general!
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