Tricurves

Click here for the full article.

Tim Lexen has a created a family of 'triangles' with curved sides that he calls tricurves. What makes tricurves visually and mathematically interesting is their abundant variety and the fact that all sides have the same curvature thus providing an opportunity for various tessellations. There are magnificent mosaics, or tilings, if you prefer, that continue to amaze us.

Other types of tricurves, such as those derived from the Reuleaux triangle and deltoid, will also tessellate, albeit in groups of four tiles. Tessellation with fewer than four different tiles has not yet been demonstrated and is open for discovery.

A proof could be very interesting!

Our Figure 1 shows a 30-60-90 Tricurve tiling .

Tim Lexen observes that shapes based on equilateral triangles seem to be able to tile in pairs, by for instance, adjoining the all-convex and all-concave shapes to form a repeated unit.

The Reuleux Triangle morphs into a concave horn.
sampler


Figure 6 from the full article for 30-60-90.	Figure 7 from the full article for 60-120-180.

Areas in Tricurves

The key part is the segment, i.e., the area bounded by a chord and its associated arc. This segment area is the difference between the sector area and the triangular area bounded by the chord and the two radial lines. This is shown below for a 90 degree angle/arc.

area

References

[1] Bourke, P., Tiling with Tricurves (2017). < http://paulbourke.net/geometry/tricurves>

[2] Garcia, L. and Nelson, M. Curves of Constant Width and Reuleaux Polygons (2007).
< ../reu/reuleaux.htm >

[3] Garcia, L., More of Reuleaux Polygons (2005)
< ../reu2/reu2.htm >

[4] Reuleaux, F., The Kinematics of Machinery, trans. A. Kennedy, Dover, 1963 (reprint of 1876 translation of 1872 original in German.)

Other spiral Deposits in the NCB:
< ..//spiral/spiral.htm >
< ..//log/log.htm >

2018