Three triangles tangram

The three triangle tangram (also known as Brügner's triangles ) was developed by the mathematician Georg Brügner († 1998). This is a special tangram that consists of only three similar right triangles from which a maximum number of convex figures can be formed.

In contrast to other tangrams, whose purpose is more of a playful nature, the three triangle tangram is based on a mathematical background.

Structure of the tangram

Structure of the tangram

When the tangram is formed as in the first picture, three right-angled triangles with a total of six different sides are created (which are marked with to in the second picture ). These triangles are similar to one another because of the equality of the interior angles . ${\displaystyle a}$ ${\displaystyle f}$

In the special case the triangles only consist of five different sides. This case occurs exactly when the diagonal to the side is in the ratio of the golden ratio , that is, when ${\displaystyle f=c}$ ${\displaystyle a}$ ${\displaystyle c}$ ${\displaystyle \Phi }$

Aspect ratios in tangram

${\displaystyle \sin {\alpha }={\frac {c}{a}}={\frac {1}{\Phi }}\approx 0{,}618}$.

The following then results for the associated angle ${\displaystyle \alpha }$

${\displaystyle \alpha \approx 38{,}17^{\circ }.}$

A peculiarity of this angle is also the relationship from which it follows. For the aspect ratios of the starting rectangle then follows ${\displaystyle \cos ^{2}{\alpha }=\sin {\alpha }}$ ${\displaystyle \tan {\alpha }=\cos {\alpha }}$

${\displaystyle {\frac {c}{b}}=\tan {\alpha }=\cos {\alpha }={\sqrt {\sin {\alpha }}}\approx 0{,}786.}$

Only with this aspect ratio in the starting rectangle does the three-triangle tangram arise, since in this case the number of convex figures becomes maximum.

The figures

The 16 possible convex figures

Because the triangles only have five different sides, there are more possibilities to put them back together to form figures.

The following 16 different convex figures can be formed:

• two rectangles ,
• two isosceles triangles ,
• two parallelograms ,
• three trapezoids ,
• two dragon squares ,
• a general square ,
• four pentagons .

Individual evidence

1. ↑ George Brügner: Three-triangle-tangram . In: BIT Numerical Mathematics . September 24, 1984, pp. 380-382. doi : 10.1007 / BF02136037 .

Web links

• General information on tangrams with a picture of the three-triangle tangram (Tangram de Brugner)
• Flash application in which you can put all 16 shapes together (SWF file; 51 kB)
• On-line Application: Three Triangle Tangrams