This is a a quick tutorial on Matchstick Construction. In it I describe how one can use match stick construction to create all sorts of geometric shapes.The official definition of Matchstick construction is, “Every point which can be constructed with a straightedge and compass, and no other points, can be constructed using identical matchsticks (i.e., identical movable line segments).” (1) There are four postulates that apply to Matchstick Construction. The first is, “A line may be laid to pass through a given point, or with one extremity at a given point.”(3) The second is, “a line may be laid to pass through two given points, or with one extremity at a given point and passing through a second given point - but the two given points may not be such as lie in a given or laid line. Note.-- The express barring of line “overlapping” is adopted to improve the elegance of the construction.”(3) Thirdly, “A line may be laid with one extremity at a given point and its other extremity on a given line.”(3) Lastly, “Two lines may be laid simultaneously to form the sides of an isosceles triangle, two of their extremities coinciding and the other two being at given points.”(3) I will illustrate how these postulates may be used, experimenting with various shapes.One thing to realize during the construction of these shapes is that the ends must meet firmly. To do this and not disturb the rest of the construction requires much dexterity of hand, but it can be done.
The use of matchsticks in Matchstick construction is a way to visually explain an abstract concept. In no way will the matchsticks be all the same length or be perfectly smooth, and substanceless things. No such “thing” exists in the universe. If this doesn’t make sense, then think back to the definition of a
Kocher 2
line that we all know, “In geometry a line:, is straight (no curves), has no thickness, and extends in both directions without end (infinitely). If it does have ends it is called a "Line Segment".(2)
The matchsticks are only a way to see and understand this abstract and interesting concept of line segments.
So you ask, “Can the line segments be of any length?” Yes, they may, but there is a stipulation. The line segments must be all the same length. This will be explained in a bit when I talk about making perfect angles and regular shapes.
The simplest true shape to make is an equilateral triangle. To do this a geometer would merely take three equal moveable line segments and have the extremities intersect.
We know that if a triangle has equal sides then it has equal angles and this is evident in the picture (figure 1) to the left. The next simplest true shape to make is a rhombus/diamond/parallelogram, constructed from two equiangular triangles that share a base as shown (figure 2). The reader will kindly notice, that the diagonal, which is equal to the sides, bisects the one hundred and twenty-degree angle, thus proving that this is a true parallelogram. This is so very cool!