Welcome sa part 2 ng unang lesson natin, Patterns and Numbers in Nature and the World. Patterns indicate a sense of structure and organization that it seems only humans are capable of producing this intricate, creative, and amazing formation. It is from this perspective that some people see an intelligent design in the way that nature forms.
The first pattern that we're going to discuss is about symmetry. Symmetry indicates that you can draw an imaginary line across an object and the resulting parts are mirror images of each other. Examples, the butterfly, Leonardo da Vinci's Vertuvian Man, and the starfish.
Isa sa mga halimbawa nito in nature ay yung butterfly. The butterfly is symmetric about the axis indicated by the line. Note that the left and the right portion are exactly the same.
So this type of symmetry is called bilateral symmetry. Next is... The Leonardo da Vinci Pertuvian Man.
It shows the proportion and symmetry of the human body. Ito yung kapag tinignan mo ang sarili mo sa harap ng salamin, yung nakikita mo sa left side ay nakikita mo rin sa right side. There are other types of symmetry depending on the number of sides or faces that are symmetrical. Isang halimbawa dyan ay ang starfish. Dahil ang starfish ay merong five-fold symmetry.
Note that if you rotate the starfish, you can still achieve the same appearance as the original position. So ang tawag natin doon is rotational symmetry. The smallest measure of an angle that a figure can be rotated while still preserving the original position, is called the angle of rotation. A more common way of describing rotational symmetry is by order of rotation.
In order of rotation, a figure has a rotational symmetry of order n times the n-fold rotational symmetry if If 1 over n of a complete turn leaves the figure unchanged. To compute for the angle of rotation, we can use this formula. 360 degree over n. So isang halimbawa nga nito ay the snowflake.
This pattern, the pattern of the snowflake repeats 6 times indicating that there is a six-fold symmetry. So, using the formula over n, the angle of rotation is Although, many combination and complex shape of snowflakes may occur which leads some people to think that no two are alike dahil maraming snowflakes na hindi perfectly symmetric because of humidity, and temperature. Another marvel of nature design is the structure and shape of the honeycomb.
Why best use hexagon in making honeycomb and not any other polygons? To conclude, why hexagonal formation are more optimal in making use of the available space Meron tayong tinatawag na packing problem. So ano ba yung packing problem? It involves finding the optimum method of filling up a given space such as a cubic or a spherical container.
So dito, meron tayong dalawang gagawin para i-compare. Para ma-prove that the hexagonal formations are more optimal in making use of the available space. So, proof. Suppose you have circle of radius 1 cm. Each of which will then have an area of pi square centimeter.
We are then going to fill a plane with this circle using square packing and hexagonal packing. So unahin nga natin dyan yung square packing. So ito yung una natin.
So ang gagawin natin para makumpayin natin, ikukumpit natin yung percentage ng square packing and hexagonal packing. To compute the percentage ng square area covered by a circle, so mula sa figure na to, itong gagamitin natin formula. Area of the circles divided by area of the square times 100%. So mula sa figure na to, meron sa bawat square, meron lang isang circle. So pag pinagsama natin itong portion na to, saka yung portion na to, At itong portion na yan ay makakabuo tayo ng isang circle.
Kaya, yung area ng circle is a pi square centimeter. And then, sa bawat square, meron tayong apat na square centimeter. So, itidivide lang natin times 100 and that is equivalent to 78.54%. So, ito yung percentage ng square packing.
Next natin is the hexagonal packing. Now, to compute the percentage, kunin muna natin yung area ng bawat triangle. For hexagonal packing, we can think of its hexagon as composed of 6 equilateral triangle with side equal to 2 cm.
So, This formula, the side squared times square root of 3 over 4. So, again, meron tayong dalawang centimeter. So, substitute lang natin. 2 centimeter squared times square root of 3 over 4. And 2 squared, 2 centimeter squared is 4 squared centimeter.
So, makakancel natin ito. So, therefore, the area of each triangle in the hexagonal. Packing is square root of 3 square centimeter. To compute the percentage ng hexagon packing, gagamitin natin formula.
So, the area of a hexagon is 6 square root of 3 square centimeter. Bakit? Dahil sa bawat triangle, merong square root of 3 square centimeter. So times 6 lang natin kaya naging 6 square root of 3 square centimeter.
And yung area ng hexagon, so yung area ng hexagon natin, this is 6 square root of 3 square centimeter. Ngayon, dahil meron tayong tatlong circle ang kasha sa loob ng hexagon, it gives a total of 3. pi square centimeter. So, ito yung isang circle. So, ito, pag pinagsama natin yan, makakabot tayo ng dalawa pang circles.
So, using the formula, area of the circles divided by the area of the hexagon times 100%, substitute lang natin, times 100%, the percentage of hexagonal packing is 90.69%. Now, Comparing the two percentage, we can clearly see that using the hexagon will cover a larger area when using square. Therefore, we can conclude that hexagonal formation are more optimal in making use of the available space. Now you know kung bakit hexagonal ang bahay ng...