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Mathematical figure in Filipino culture (<strong>Star</strong>)High school GeometryDuring Christmas season, Filipino people decorate rooms,houses and any building with various objects such as a star,which is shown on the right. Since it is said that Jesus Christwas born guided by a star, since then, the star has become o<strong>ne</strong>of the symbols of Christmas.As you notice, the shape of the center of a star is a regularpentagon. If you con<strong>ne</strong>ct all vertices of a star <strong>to</strong> the <strong>ne</strong>xtvertex, you can also form a big regular pentagon.This fact is the big hint when you <strong>construct</strong> a star on thepaper accurately.In other words, if you can <strong>construct</strong> aregular pentagon, you can also <strong>construct</strong> a“star” by only drawing diagonal li<strong>ne</strong>s.♦Objective of this lesson♦To be able <strong>to</strong> <strong>construct</strong> a star mathematicallyLet us review the characteristics of a regular pentagon before we start <strong>to</strong> <strong>construct</strong> it.Q1: <strong>How</strong> many sides are there? 5Q2: <strong>How</strong> many vertices are there? 5Q3: <strong>How</strong> many diagonal li<strong>ne</strong>s can we draw? 5Q4: Are the lengths of diagonal li<strong>ne</strong>s equal <strong>to</strong> each other?Yes. We can form a star if we have five equal size sticks.But our <strong>to</strong>day’s objective is <strong>to</strong> <strong>construct</strong> a star “on the paper”. Since we can’t move thediagonal li<strong>ne</strong>s on the paper, we have <strong>to</strong> come up with another idea <strong>to</strong> <strong>construct</strong> it.Let’s check the measure of angles of a regular pentagon <strong>ne</strong>xt as follows.Q5: What is the sum of the measure of interior angles? 540°
Q6: What is the measure of an interior angle of a polygon?The formula for the sum of interior angles of polygon whose number of sides is n is180°(n-2). When n=5, 180°(5-2)=180°(3)=540°. 540°/5=108°.Q7: What is the measure of an interior angle of a star?108°÷3 = 36°Since an interior angle of a star is 36°, it’s very difficult <strong>to</strong> <strong>construct</strong> a star without the useof a protrac<strong>to</strong>r.But we are not allowed <strong>to</strong> use a protrac<strong>to</strong>r in <strong>construct</strong>ing. We are allowed <strong>to</strong> use only acompass and a straightedge.Do you think that a star is a beautiful figure?The hint <strong>to</strong> <strong>construct</strong> a star exists in the beautyof it.Namely, it is “The Golden Ratio”.Let us introduce “The Golden Rectangle (Ratio, Section)” <strong>to</strong> you!This can be the hint in order <strong>to</strong> <strong>construct</strong> a star.If you have a cell card, please look at the shape ofit carefully. As you see, the shape of a cell card is arectangle. The measure of sizes are as shown in thepicture on the right.Likewise, the ratio of width <strong>to</strong> height of cash cardand calling card is also close <strong>to</strong> 1: 1.6. We can getthis ratio from the proportion as shown below.5.35cm1 : x = x : ( 1 + x ) (x>1)2x − x −1= 01+ 5x =28 .5 ÷ 5.35 = 1.598.5cmThe ratio of a shorter length <strong>to</strong>a longer length is equal <strong>to</strong> alonger length <strong>to</strong> a whole length.This ratio “1:1.618” is called “the golden ratio”. It has enchanted peoples’aesthetic sense by the beauty of its balance beyond the centuries. The golden ratio isalso seen <strong>to</strong> the pyramid in Egypt and pictures drawn by Leonardo Da Vinci.In this page, we introduced only the summary of “The golden ratio”. We suggestyou <strong>to</strong> research it more. It is worth studying.
<strong>How</strong> <strong>to</strong> obtain the golden section (ratio) with the use of a compass and a straightedge?ADAMDAMD111BV1CB12N12CBN12CConstruct any square,ABCD○1○2 Bisect the squarewith segment MN○3 Draw a li<strong>ne</strong> segment NDA122 ⎛ 1 ⎞1 + ⎜ ⎟⎝ 2 ⎠=54M=52DF○4 Extend side AD○5 Using a compass, make arc EDusing center N and radius DN .BNC1 5 1 5+ =+2 2 2E○6 Extend side BC until it intersectsthe arc at point E○7 Construct segment EF perpendicular <strong>to</strong>segment BE , and side AD intersects sideEF at point F. Then ABEF is a golden rectangle.→Back <strong>to</strong> the problem<strong>How</strong> <strong>to</strong> <strong>construct</strong> a star by utilizing golden rectangle?1Golden rectangle1.618Construct a golden rectangleSet the compass adjusting the radius <strong>to</strong>equal with the length of a goldent l
Draw a li<strong>ne</strong> segment. Then fromthe endpoint of it, create an arcFrom the same point, createan arc on the upper left sideFrom the intersection, createan arc on the upper right sideSet the compass adjustingthe radius <strong>to</strong> equal with thewidth of a golden rectangleFrom the end point, createan arc on the upper right sideFrom the intersection, createan arc on the upper left sideCon<strong>ne</strong>ct the <strong>ne</strong>wly created points and the li<strong>ne</strong> segmentFrom the <strong>ne</strong>wly createdpoints, create an arc on theupper left sideFrom the <strong>ne</strong>wly createdpoints, create an arc on theupper right sideCon<strong>ne</strong>ct the <strong>ne</strong>wly createdpoint <strong>to</strong> other points <strong>to</strong> beable <strong>to</strong> create a pentagonDraw 5 diagonal li<strong>ne</strong>sAte<strong>ne</strong>o De Davao University – Regional Science Teaching Center
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