Dividing a segment into golden ratio.
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Tema: Basic geometric shapes
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Enunciado
The golden number might have been the first irrational number known to the Greeks. When the Pythagoreans discovered that irrational numbers existed, i.e. that they could not be written as the quotient of two whole numbers, they were dismayed, as this fact broke many of their philosophical theories. That is why they decided to keep this discovery a secret.
Theano of Crotone, a Pythagorean mathematician, was the first woman to carry out these divisions, confirming thus the existence of irrational numbers. As a good Pythagorean, she believed and defended that all material objects were composed of natural numbers, so that the measure of anything could be expressed with an exact measure. However, she was also the first to posit the existence of the golden ratio as the essence of the universe.
The golden number, or golden ratio, is represented with the Greek letter Φ (Phi), honouring Phidias. Let's divide any segment into two parts a and b so that a/b is the golden ratio.
- Draw a segment with points at the ends A and B. Let's call it AB.
- Through point B, draw a perpendicular that measures the same as AB.
- On this perpendicular we find the midpoint, which we will call D, and it joins with A, forming the right-angled triangle ABD.
- With the compass, we prick on D and open it up to point B. We draw an arc that cuts the hypotenuse of the triangle. This cutting point is E.
- With the compass, we prick on A and open it up to point E. We draw a new arc that cuts the original AB segment. This last point is C.
- We have managed to divide segment AB into two parts, AC and BC, the bigger part is AC and the smaller BC, both lengths are in golden ratio.
- Measure the segments AB and AC and calculate the quotient AB/AC. Similarly, if we divide the length of the segments AC and BC we find the same number.
- In both cases the same result is obtained, the golden number: "The whole is to the part as the part is to the remainder".
$$\frac{AB}{AC} = \frac{AC}{CB}\;,\qquad \Phi = \frac{1+\sqrt{5}}{2}$$
Phi is the golden number, also known as the "divine proportion".
Observaciones y contexto
- In this guided activity, students need material to draw perpendiculars and arcs of circumference, as well as a ruler to measure the segments found. It is important to try to accompany them so that they are able to follow the instructions.
- The aim of the activity is to put into practice the knowledge learnt about perpendiculars, and triangles and to practice with rulers and compasses. The golden number is used to introduce Theano in the construction and we can mention at the end what an irrational number is. It is also interesting to note at the end of the activity that all students will have found a very similar number, different approximations of the golden number, regardless of the initial segment they have drawn.
- Crotone was in Theano's time a colony of Magna Graecia.
- Enheduanna (25th century BC) was a predecessor of Theano of Crotone, considered the first recorded woman in the history of science and the first to sign her works, in cuneiform script.
- Some of Theano's contemporaries are other women in the Pythagorean school that were born around 500 BC, such as Damo, Myia and Arignote of Crotone, considered to be daughters of Theano and Pythagoras by several authors. Even though there is not much information about them, some other women belonging to this group were Babelica of Argos, Beo of Argos, Quilonis, Echecrates of Phlius, Ecellus and Ocellus Lucanus, Habrotelia of Tarento, Cleecma, Cratesiclea, Lastenia of Arcadia, Pisirroda of Tarento, Filtis, Teadusa, Timica and Tirsenis of Sibaris.
- After Theano we can mention Aglaonice, or Aganice of Thessaly, (3rd century BC, known for her ability to predict eclipses) and Hypatia (4th century AD).
Descripción
Ruler and compass exercise. The aim of the task is to divide a segment into golden ratio through a step-to-step guided ruler and compass construction.