The Magic of Numbers – Serial 10

Sensing shapes

From the moment we are born, we simply felt our size via the sense of touch, from the sense of touch we feel the other parts of our body. The scientific study of shapes is called geometry.

Then we saw that everything has a shape with our sense of sight and touch and we started to compare everything with the size of our body. Then we remembered all of the sizes via our human memory and that gave us the ability to recognize objects.

When you are already a teenager, sometimes you are reminded of the place you grew up as a toddler. When you revisit that place as a teenager, it feels smaller compared to you toddler years. You are reminded of how the sights and sounds of the place you grew up.

Defining the basic terms of geometry based on how we feel about a place

We simply know shapes without needing to define the words point, straight line, angle, area and volume. We simply know from our sense of sight and touch. We record what we detect to our memory. Our brain simply processes the positions of all the objects our eye can detect.

When we look at an object we may see a sharp edge, in geometry, we call that a point. We may also call a point the place when the two walls of house intersect.

We know we can scratch the sand on the beach, and it leaves a trail, we can call that trail a line, it can be straight or crooked. We also leave markings when use a pen to write on paper.

As you walk around your room it may feel wide, narrow or spacious. This created the feeling of area. We feel area when we walk around a place.

Not only that, you can also jump happily in your bed, but you take care, as a survival instinct, not to get your head too close to the ceiling as you jump. Or if you have a pet parrot, it can fly from the bottom of your bookshelf to the ceiling. This how we feel the room has volume.

The higher the ceiling, better it feels, or to avoid the ceiling, you jump happily on the trampoline in your backyard instead. We can also feel volume when we we can hold the objects and feel the size (thickness) with our sense of touch.

When we feel we cannot jump in our bed because the ceiling is too low compared to the height of our body, or can only take a few walking steps in the place, or if there are too many objects or people in a room, we might feel claustrophobic (fear of enclosed spaces).

This is how we compare the size of the room to the size of our body without measuring instruments, using only the sense of sight and touch, along with our memory.

The difference between length, width and height

In Serial 7 we were not concerned with the width of thickness of an objects, but only length and height. How we define length and height is based on our eye sight and our ability to define what is vertical and horizontal.

When we can move our hands left and right based on our vision field, or we can walk around the place, we feel something is horizontal. The spacing from the trail from left to right can be called length. When something can lay by itself on the ground, like a sleeping hamster, dog or cat, it can be called horizontal.

We we see an object falling into the ground, like raindrops and snowflakes, we feel something is moving from up to down, this is called horizontal. We can call the trail of the snowflake from the top of the oak tree to the bottom of the oak tree as height.

When we can hug with our arms an oak tree in the park or the banana tree in our backyard, we feel the tree has thickness, sometimes this is called width. We can also feel the thickness of an object when we grasp it with our hand, like a banana.

But length, width and height are also distances. The words width and height were only created to differentiate from the word length, because they have different positions.So they are simply measurements and we can call them the spatial or geometric dimensions. In engineering, the word dimension and measurement is sometimes used interchangeably, thus creating semantic conflicts.

What is an angle?

If we nail a slender and straight pice of paper into a square soft cork plate, we know that the piece of paper can be rotated from where it is nailed at. But to know whether it is rotating, we have to mark a reference point, say an orange in the table. We can also use our memory as a reference point because we are good at remembering the positions of objects.

Here we can say that the slender piece of paper has rotated. If it goes left, it is called anti-clockwise. If it goes right, as shown below, it has went clockwise.

Now you return the slender paper to its original position and you apply a water-color underneath the free end. Then you rotate it again: You can see that it forms a circle as you fully rotate it. This is called as going one round, or one lap, a common term used in racing.

The yellow part is the circumference of the circle, where the water-color has swept across. The slender piece of paper is the radius of the circle. Two radiuses make one diameter.

Now we will split the circle into 8 equal parts (fractions), to better keep track of how far we have been from the reference point, and we use a different color for each slice. Just like we can use the tiles in our house to keep track of how far we have been.

We can say we have moved four tiles from our starting point. If one tile is has the length of one foot, we can also say we have moved four feet.

Here we have moved two slices from the reference point. This can be called as a right angle. A right angle can be called a quadrant, because we have moved two quarters. We also know that no matter how big or small the circle, we will move the number of slices. When things, for example, are standing, we call them as perpendicular (right angles) to the ground.

The next picture is an obtuse angle, because we have moved more than two slices from the reference point. When we move one slice only that is an acute angle. We can now see that angles were derived from how many slices the paper has moved from a reference point. Angles are used to measure how far two lines intersect.

To measure angles more precisely, authorities have decided to split the circle into 360 equal parts. Each slice of that is called a “degree”. So a right angle is 90 degrees if a circle is cut into 360 equal parts. A “degree” is smaller than the slices we have used in this serial. Angles are very important in determining geometric properties.

Now detaching the slender piece of paper form the cork, we now put two lines in our circle. We remove the orange as our reference point. Our reference point is now the blue line. The intersection between the blue line and the green is 2 slices or 90 degrees (if we use a 360 slice circle). This is an important geometric property of squares and rectangles. Studying geometry will never make sense without angles.

We can also say that the green line is two slices away from the blue line, likewise, we can also say the blue line is two slices away from the green line.

Creating shapes and geometrical figures

Now that the angle is defined, it is now easy to describe how we create shapes. The square, rectangle, rhombus and the cube (solid) will only be described in this serial. Some shapes (e.g a rock from the park) are not definite (does not obey geometrical measurements), and these shapes are called amorphous.

How do we create squares? We will use a piece of paper and a pencil to create marks in the paper. We make sure the all of the lines are straight, with the length and width are the same. It has no height because it is we can only see it from a flat surface. We make sure the sizes of the intersections of the lines are the same and they are perpendicular. there are 4 line intersections.

How do we create rectangles? Same method as the square except that the either the length or height has to be different.

How do we create rhombuses? All of the lines have to the same length, except that there have to be two opposite acute angles and two opposite obtuse angles.

How do we create a cube? When we add a height to some something that already has a length and width. This happens when we include a height after drawing the square in the paper. The height should equal the length and width in the paper and should be perpendicular to the plane of the paper. We can make the cube out of play-dough.

A cuboid can made if the length, width and height are different but the height should be perpendicular to the plane of the paper.

Case study: Creating simple chairs

We use our knowledge of geometry to create the solid and column. Now that we know square and cuboids are simple geometrical figures, we can now design a simple chair without a backrest. Gravity acts on the solid, but the column or legs of the chair resists the gravity acting.

We need at least 5 cuboids. 4 tall cuboids for the legs and one cuboid to be sat on (its height shorter than the taller cuboids but the length and width larger than the taller cuboids). Here is the front view of the chair.

Based on the average size of a person, we will set the measurements for the cuboids. The simplified ratios of the length width and height is also given:

We will also select the best materials for our design, we will try wood, rubber and modeling clay. Only wood is the most resilient, the other materials fail, all with the same measurements.

We need a volunteer to sit on a chair we will try two volunteers: an regular adult and a 10-year old kid. The ratios are important if we want to scale down the chair to be more comfortable for a child. This is also best when we want to design chairs for infants. This is a chair with all measurements halved, the ratios stay the same:

Just like the ratio of the circumference of the circle above to its diameter will always stay the same, no matter how big the circle is, as big as your park or as big as your computer. We gave a name to the ratio, it is called “pi” (π, a Greek character.

Now that we have wood as the best material, the next time we create chairs, they will be made out of wood. This is how we create the chairs out of trial and error. We also know that the chair will fail if we stack a lot of heavy objects on it, or if the legs of the chair are too thin. We also know if adults sit on the chair for infants, the chair will break.

Complicated things always start from the simple ones such as this. Without having this basic knowledge of material strength, ratios and geometry, the buildings we take for granted could not have been possible.

By using mathematical formulae (algebra), we reduce significantly trial and error approaches, like the material tests. The inconvenience of trial and error approaches inspired the creation of mathematical formulae in engineering applications by doing experiments. In engineering, the legs of the chair can be called columns. Columns are the basic component of buildings.

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