Vecten's Mesh

Vecten's configuration consists of a triangle with squares on its sides very much like a more common Bride's Square formed for right triangles. There are grounds to believe that already Euclid in his Data (that preceded the Elements) was interested in this configuration, but, for example, R. Simson's rendition of Data does not mention the configuration that we now credit to M. Vecten.

Vecten's configuration may be thought of as a starting element of a sequence wherein one keeps joining the outer vertices of adjacent squares and forming additional layer of squares on top of them. The extended configuration is also sometimes referred to as Vecten's. F. van Lamoen has studied a generalization that begins with a hexagon and called the added layers formed by squares and auxiliary quadrilaterals "wreaths". There were also online discussions, for example, at the Ask NRICH forums. I chose Vecten's Mesh as the title of this page to distinguish it from the original configuration.

The applet below illustrates some of the properties of Vecten's mesh.

First of all, starting with the second layer the wreaths consist of three squares and three trapezoids of equal areas. For the first layer, the trapezoids degenerate into triangles known as flank triangles of the base one. All four have the same area. To see how this feature propagates one layer further, draw the diagonals (QDiagonals in the applet) of the quadrilaterals (claimed to be trapezoids). The diagonals cut off the quadrilaterals triangles that are the flanks of the triangles on the previous level. In each quadrilateral there are two such triangles and these share a base and have the same area implying that their third vertices lie on a parallel to the base. Thus the quadrilaterals are indeed trapezoids. Further extension is covered by F. van Lamoen. The sides of the squares are parallel and perpendicular to the sides or to the medians of the base triangle according as whether the layer is odd or even numbered. The midlines of the squares in a wreath form a triangle homothetic to the base triangle. All such triangles are then homothetic to each other. The applet shows that the lines joining the centers of opposite squares in the two last layers are concurrent. In the applet those lines are referred to as F. van Lamoen's. (SDiagonals are the diagonals in the squares and help identify Lamoen's lines.)

References

1. S. Johnston-Wilder, J, Mason, Developing Thinking in Geometry, The Open University, 2006
2. F. van Lamoen, Square Wreaths Around Hexagons, Forum Geometricorum, Volume 6 (2006) 311–325.
3. R. Simson, The Elements of Euclid (also Euclid's Data), Elibron Classics, 2005

Copyright © 1996-2008 Alexander Bogomolny