Geometry in the open air

Yakov Perelman

Geometry

in

the open air

Science

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Printed in United States.

ISBN : 9782917260463

10   9   8   7   6   5   4   3   2   1

Table of Contents

CHAPTER ONE

GEOMETRY OF THE FOREST

CHAPTER TWO

GEOMETRY OF THE RIVER

CHAPTER THREE

OUTDOOR TRIGONOMETRY WITHOUT FORMULAS AND TABLES

CHAPTER FOUR

CHAPTER FIVE

GEOMETRY OF THE ROBINSONS

CHAPTER SIX

GEOMETRY IN THE DARK

CHAPTER SEVEN

OLD AND NEW STORIES ABOUT THE CIRCLE

CHAPTER EIGHT

CHAPTER NINE

CHAPTER TEN

CHAPTER ONE

GEOMETRY OF THE FOREST

Measuring height using shadow

I am still amazed when I think about the first time I saw a gray-haired forester, standing near a huge pine tree, measuring its height using a small pocket device. I was expecting the old man would begin to climb up the tree using a rope. Instead, he aimed his square device to the top of the tree and then he put device back in his pocket and declared that the measurement has been completed. And I thought it has not yet begun...

I was young at that time and unfamiliar with geometry and mathematics. I thought that measuring a tree without having to cut it or climb to its top was something close to a miracle. But later, when I started to study geometry, I understood how such miracles were simple applications of geometry notions. There are plenty different ways to produce similar measurements using quite simple devices that did not require any fixtures.

The easiest and most ancient way is, without a doubt, the one that has been used by Greek sage Thales six centuries BC in Egypt to measure the height of a pyramid. He used the shadow of this one. The priests and the pharaoh, gathered at the foot of the highest pyramid, were puzzled to see the northern stranger guessing the height of the huge pyramid using its shadow. Thales – the story goes – chose a day and an hour when the length of his own shadow was equal to his height; at that same time, the height of the pyramid should also be equal to the length of its shadow.[1] That is perhaps the only case where a person has benefited from his shadow...

The above idea of the Greek sage seems to us childishly simple now, but do not forget that we look at it from the point of view of all the geometric concepts that have been created since then. Thales lived long before Euclid’s wonderful book about geometry. Thales didn’t know all these geometry concepts that are now known by every schoolboy. However, using the shadow to calculate the height of a pyramid implies some knowledge of the geometric properties of a triangle, and in particular the following two properties (the first was found by Thales himself):

Two angles of an isosceles triangle are equal, and vice versa – the sides that are opposite to equal angles of a triangle are equal.

The sum of angles of any triangle is equal to two right angles.

Armed with this knowledge, Thales could conclude that, when his own shadow is equal to its height, the Sun’s rays meet the flat ground at an angle that is half right, and consequently, the top of the pyramid, the middle of its based and the extreme point of its shadow must designate an isosceles triangle.

This simple method is very convenient; it could be easily used to measure lonely standing trees whose shadows do not merge with neighboring shadows. But in our USSR latitudes, the process is not as easy as in Egypt. Indeed our Sun is very low above the horizon, and the shadows are equal to the height of their items only during mid-days in summer months. Therefore, the method of Thales in this form is not always applicable.

It is, however, easy to change this method so it would be applicable on any sunny day without the need for the shadow to be equal to the object. You simply need to measure the shadow of a stick and then the shadow of the object (a tree for example) (Figure 1), and calculate the height of the object using the following equality:

AB : ab = BC : bc,

length of the tree shadow and the length of the stick shadow. This obviously follows from the geometric similarity of triangles ABC and abc (two equal angles).

Some readers may object, perhaps, that such a basic method does not need a geometric justification. Indeed, shouldn’t the ratios between the heights of objects and the lengths of their shadows be equal? The point, however, is not as easy as it seems. For example, you cannot use this method to the shadows cast by the light of a street lantern. In the Figure 2 you see that the column AB is approximately 3 times higher than curbstone ab, however, the shadow of the column is eight times longer than the shadow of the curbstone. Explain why in one case the method is applicable and why in the other it is not. This explanation cannot be made without geometry.

Figure 1: Measuring the height of a tree using shadow

Problem

Take a closer look, what is the difference between the Sun beams and the lantern’s one? The main difference is that the former are parallel to each other while the latter are not. This is clearly evident with the lantern’s beams; but why it is correct to consider the Sun beams as parallel, even if they original from the same point?

Figure 2: When such measurement is impossible

Solution

When the sunlight reaches the Earth, we can assume that the Sun’s rays are parallel because the angle between them is extremely small, almost imperceptible. A simple geometric calculation will convince you of that. Imagine two rays emanating from any point of the Sun and reaching the Earth at two points that are, let’s say, one kilometer from each other. So, if we put a compass leg at the point of the Sun, and have the other leg describing a circle whose radius is equal to the distance from the Sun to the Earth (i.e. a radius of 150 million kilometers), then the length of the arc formed by the two rays would be 1 kilometer, while the full circumference of this gigantic circle would be equal to 2∙π∙150 million km = 940,000,000 km. One degree of this circle (1/360 times) would represent about 2.6 million kilometers; one arc minute is 60 times less than a degree, i.e. is 43,000 km, while one second arc is 60 times less than an arc minute, i.e. 720 km. But our arc is just 1 kilometer long, therefore, it represents just 1/720 of an arc second. Such an insignificant angle is elusive even for the most precise astronomical instruments; therefore, in practice, we can consider the Sun’s rays as parallel lines when they reach the Earth.[2]

If these geometric considerations were not known to us, we could not reasonably have used the method based on shadows to measure heights of objects.

However, in practice trying to apply this shadows method to measure heights does not yield reliable results. The shadows are not clearly demarcated to measure their length in an accurate manner. The shadows cast by the light of the Sun have vaguely-defined borders which create uncertainty about their edges. This occurs because the Sun is not a point; instead it is a large luminous body that emits rays from many points. Figure 3 shows why the shadow of tree BC has another appendage CD that gradually taper off. The error originating from the fact that the shadow measurement is not very accurate can reach up to 5% and even more if the Sun is not too low standing. This error is added to the other inevitable errors – from surface irregularities, etc – and greatly reduces the accuracy of the resulting measure. In mountainous areas, for example, this method is completely inapplicable.

Figure 3: Explanation for the vaguely defined shadow

Using the method of Jules Verne

Next is a - also very simple - method of measuring the height of tall objects. This one was picturesquely described in Jules Verne’s famous novel The Mysterious Island.

"Today, we need to measure the height of the granite wall, - said the engineer.

- You will need this tool? - Herbert said.

- No, it not required. We will act somewhat differently, referring to a simple and accurate method.

The young man, trying to learn as much as possible, followed the engineer who came down from the granite wall to the edge of the shore.

The engineer used a straight stick, 10 feet long, as a reference to which the height of wall would be measured. He also carried with him a plumb line which is nothing more than a stone tied to the end of a tope.

The engineer moved approximately 500 feet away from the granite wall, and then stuck the stick vertically in the stand and firmly cemented it. He made sure this one is absolutely vertical using the plumb line.

Then he walked slightly away from the stick at such a distance that lying on the sand, it was possible to see the top of the stick and the top of the wall in one straight line (figure 4). He carefully marked that point on the sand using a milestone.

- Do you know the rudiments of geometry? - He asked Herbert, while getting up from the ground.

- yes.

- Remember the properties of similar triangles? - Their congruent sides are proportional.

- Right. So, now I will build two similar right triangles. The smaller one is built with the stick, and the milestone, while the other is built using the wall and the milestones. These two triangles are rights and they have one equal angle (the one emanating from the milestone). Consequently these two triangles are similar.

- Got it! - Exclaimed the young man. – The ratio between the distance of the milestone to the stick and the distance of the milestone to the wall is equal to the ratio between the height of the stick and the height of the wall.

Figure 4: How to measure the height of a wall using Jules Verne’s method

- Yes. And therefore, if we measure the first two distances, then knowing the height of the stick, we can calculate the fourth unknown member of the equality, i.e. the wall height. We will do so without direct measurement of the height.

Both horizontal distances were measured: the distance of the milestone from the stick was 15 feet while the distance of the milestone from the wall was 500 feet. After the measurements, the engineer made the following entry:

15 : 500= 10 : x,

500 * 10 = 5000,

5000 : 15 =333.3.

The granite wall height was equal to 333 feet."

How the Sergeant did it

Some of the above methods used to measure the height are inconvenient as you may need to lie on the ground. You can, of course, avoid such inconvenience. This is just what the Great Patriotic War. Lieutenant Ivaniuk did. His unit was ordered to build a bridge over a mountain river. Germans were entrenched on the opposite shore. Lieutenant assigned a reconnaissance group for the survey of the site where the bridge would be constructed. The survey team was headed by Senior Sergeant Popov who measured in the near woodland the diameter and height of the most common trees and counted the number of trees that could be used for construction.

A tree height was determined using a pole (pylon) as shown in Figure 5.

Figure 5: Measuring the height of a tree using a pole

This method can be summarized as follow:

Use a pole and plug it vertically into the ground at a distance from the tree to be measured (Figure 5). Step away from the pole to a point A, from which, looking at the top of the tree, you’ll see a straight line passing by the top point b of the pole. Then, without changing the position of the head and looking in the direction of the horizontal line aC, note points c and C, in which the line of sight meets the pole and the tree trunk respectively. Ask a colleague to tag points A, c, and C. Now, using the similarity of triangles abc and aBC, you can calculate the distance BC.

BC : bc =aC : ac,

Whence

BC = bc∙(aC/ac)

The distances bc, aC and ac are easily and directly measured. However in order to find the total height of the tree, the distance CD must be added to BC. CD can be measured directly.

To determine the number of trees, the sergeant ordered the soldiers to measure the area of the forest. He then calculated the number of trees in a small area the size of 50 m * 50 m and produced the corresponding multiplication.

Based on all the collected data, the unit commander established the area where the trees need to be cut down. The bridge was built by the deadline and the combat mission was successful.[3]

Without getting close to a tree

It happens that for some reason it is difficult to get close to the base of the tree to be measured. How can we determine its height in this case?

This is possible using an ingenious device that is close to the previous ones and that can be easily made by the observer. Two sticks ab and cd (see Figure 6) are fastened at right angles so that ab and bc are equal while bd is half ab. That is the whole device. To measure a height, hold the device in your hands, hold the stick cd vertically (you can associate it with a plumb line and a small weight). You need to repeat this process at two different locations as shown in figure 6: First at point A, where the device is place with point c on the top, and then at point A’, where point d is placed on the top.

Figure 6: Using a simple altimeter, consisting of two sticks

Point A is selected so that, looking along ac, we see it aligned with the top of the tree. Similarly, point A’ is selected so that looking along a’d’, we see it aligned with the top of the tree. When searching for the two points A and A’[4], we are in fact measuring the tree, because the desired height BC of the tree is equal to the distance AA’. This equality is easy to prove from the fact that aC = BC, and a’C = 2BC; this means

a’C – aC = BC.

You can see that by using this simple tool, we measure the height of tree without getting close to it (we need only to measure the distance AA’) and without any calculations.

Instead of two sticks, you can use four pins on the board by placing them properly. Such device is even easier to use.

Foresters’ altimeter

It is time now to explain how to construct the real altimeter which is used by foresters in practice. I will describe one of these altimeters. The essence of the device can be seen in figure 7. Cardboard or wooden rectangle abcd is held in a way so that, looking along the edge ab, you will see the top of the tree B.

Figure 7: Scheme use altimeter Foresters

From point b hung on a thread sinker q. Note the point n where the thread crosses the line dc. Triangles bBC and bnc are similar, since they are both right and have equal acute angles bBC and bnc (whose sides are parallel). Hence, we can write the following proportionality:

BC : nc = bC : bc;

Whence

BC = bC·(nc/bc)

Since bC, nc and bc can be measured directly, it is easy to obtain the desired height of the tree, by adding the length of the bottom of the CD trunk (which is equal to instrument height above the ground).

Some few details need to be added. It is possible to divide the side dc into divisions as the ratio nc/bc is always expressed as a decimal fraction. These divisions directly indicated the ratio nc/bc which is equal to the ratio BC/bC. So you have simply to multiply the indicated ratio on the altimeter by your distance from the three to find the height of this one. Suppose for example that the thread settled in front of the 7th division (i.e. nc = 7 cm); this means that the height of the tree above the eye is 0.7 the distance of the observer from the trunk.

Figure 8: Foresters Altimeter

The second improvement relates to the method of