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  1. Design & Illustration
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Thiết kế hình học: Những điều cơ bản

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Difficulty:BeginnerLength:LongLanguages:
This post is part of a series called Geometric Design for Beginners.
Geometric Design: Working With 4 and 8

Vietnamese (Tiếng Việt) translation by T. Art (you can also view the original English article)

Final product image
What You'll Be Creating

Trong nền văn minh trước đó, khoa học (toán đặc biệt), tôn giáo và nghệ thuật đã không riêng biệt. Chúng tôi thậm chí không có một từ cho lĩnh vực rộng và chất lỏng họ thành lập với nhau, nhưng chúng tôi có thể nhận được một cảm thấy cho nó bởi nhìn bất cứ lúc nào của các tác phẩm nhiều đáng kinh ngạc của hình học nghệ thuật đó đã sống sót cho đến bây giờ, và thường là một phần của một cấu trúc thiêng liêng.

Rose window in Notre Dame cathedral
Các cửa sổ rose Bắc tại Đại giáo đường Notre Dame, Paris.

Geometry is nothing but numbers made visible. Trong thực tế nó là biểu hiện đầu tiên của số điện thoại, trước khi viết tắt biểu tượng — 1,2,3 — được tạo ra cho họ. Đầu geometers hiểu các mối quan hệ giữa số bằng cách nhìn vào cách hình hình dạng liên quan đến nhau, và như con số có ý nghĩa sâu sắc, vì vậy đã được các mô hình đang nổi lên từ họ tính phí với ý nghĩa. Bản chất hai chiều, trừu tượng của hình học được hiểu như là một bước tiến gần hơn để Divine chiều bằng không, không thể biết hơn thế giới vật lý của chúng tôi, và vẻ đẹp của nó là khá nghĩa là ra khỏi thế giới này.

Tessellated pattern in the Alhambra
Tessellated mô hình trong Alhambra, Tây Ban Nha. Photo by Gruban.

Niềm đam mê với hình học và toán học mô hình reemerging vào ngày hôm nay: chúng tôi có thể nhìn thấy nó trong sự phổ biến ngày càng tăng của nghệ thuật fractal. Có là không cần thiết, Tuy nhiên, đối với các phần mềm đặc biệt để tạo ra thiết kế hình học rất phức tạp, và nó là trong thực tế đáp ứng sâu sắc, ngay cả suy nghi, để từ từ rút ra cho họ ra khỏi hư vô trắng của một tờ giấy, như chúng tôi sẽ làm trong những hướng dẫn.

Chúng tôi sẽ bắt đầu với các khối xây dựng của hình học, nắm vững các công trình xây dựng đơn giản trong thời gian đầu tiên vài bài học. Sau đó chúng tôi sẽ chuyển sang mô hình và công trình xây dựng phức tạp hơn, và những bài học vài cuối sẽ giải quyết tác phẩm thật sự phức tạp, nhưng bổ ích, của hình học.

Thuật ngữ

Để bắt đầu với, chúng ta hãy xác định một vài điều khoản mà sẽ đi lên thường xuyên trong những bài học. Bạn có thể đã có quen với một vài người trong số họ.

Terminology 1
  • Một vòng tròn là hình dạng hình học đơn giản nhất, một đường cong đóng cửa, nơi tất cả các điểm là cùng khoảng cách từ Trung tâm.
  • Một đường kính là bất kỳ dòng kết nối hai điểm trên một vòng tròn và chạy qua Trung tâm.
  • Bán kính là bất kỳ dòng kết nối Trung tâm của một vòng tròn với chu vi của nó (thực tế nói, đây là của chúng tôi la bàn mở khi vẽ một vòng tròn).
  • Một dây nhau là bất kỳ dòng kết nối hai điểm trên một vòng tròn, mà không đi qua Trung tâm.
  • Một hình bán nguyệt là chính xác một nửa một vòng tròn.
  • Một vòng cung là bất kỳ phân đoạn của vòng tròn đó không phải là một hình bán nguyệt.
  • Một ốp là một dòng mà chỉ cần chạm vào một vòng tròn ở một điểm duy nhất.
Terminology 2
  • Một góc cấp tính là nhỏ hơn º 90.
  • Một góc bên phải là chính xác º 90, và hình vuông nhỏ được đánh dấu bên trong nó là cách thông thường để chỉ ra một góc bên phải trên một sơ đồ.
  • Một góc obtuse là lớn hơn º 90.
  • Một tam giác là bất kỳ hình dạng khép kín với ba mặt thẳng. Một tam giác ngẫu nhiên, như trái ngược với sau ba, cũng được gọi là tam giác scalene. Tổng các góc trong tam giác bất kỳ luôn luôn là 180º.
  • Một tam giác có một góc bên phải. Hai góc độ khác không cần phải được bình đẳng, và các bên khác nhau.
  • Một tam giác isosceles có hai mặt bằng (bằng độ dài được biểu thị bằng dấu gạch ngang trên một sơ đồ).
  • Một tam giác đều có ba mặt bằng, và ba góc của nó đều bình đẳng (60º).
Terminology 3
  • Một tứ giác là bất kỳ hình dạng đóng cửa với bốn mặt thẳng. Tổng các góc trong một tứ giác luôn luôn 360º.
  • Một hình chữ nhật là một tứ giác với bốn góc. By necessity, the two sides opposite each other are parallel and the same length.
  • A square is a specialized rectangle where all four sides are equal.
  • A rhombus also has four equal sides, with the two opposite each other parallel, but no right angles.
  • The next eight shapes are polygons (closed shapes with more than four sides) with five, six and up to 12 sides. All their sides and angles are equal.

Tools

Geometry was originally practiced with nothing but a rope and pegs, so it really doesn't require fancy tools, only accurate ones, when working on paper. You only need three things: a pencil, a straight edge, and a compass.

Pencils

A basic lead pencil is perfectly adequate for the job, but don't just grab the first one you find: it needs to be the right hardness. In the picture below you'll notice the label HB on the orange pencil, and 6H on the grey one. These are indicators of hardness. B indicates a soft lead, and the higher the number (4B, 5B), the softer.

A soft lead will leave a darker mark that does not score the paper, but smudges easily. H indicates a hard lead, similarly graded, that will only leave a light mark, and not smudge, but will score the paper if pressed too hard. HB, obviously, is a happy middle.

Traditional pencils

When constructing geometric patterns, you don't want soft pencils! The reason is that dark construction lines quickly get confusing, and smudging is inevitable. Soft leads also lose sharpness very quickly, resulting in either constant sharpening, or loss of accuracy when drawing.

What we want, instead, is to build up the drawing with light construction lines, and use a softer pencil to pick out the final lines of the patterns. This is what these two pencils are for: the 6H remains sharp a long time and makes a very light line, over which final lines made by the HB really stand out.

For very complicated patterns, an intermediate darkness of line can be added in‑between, for instance a 3H or 2H. It is important, however, to learn to draw lightly with H pencils, because they do score the paper and that's a mark that can't be rubbed out. When penciling is complete, the pattern can either be inked, and the pencil then rubbed out, or painted, which will cover the pencil, or transferred to a completely clean sheet of paper using tracing paper if so desired.

The advantage of traditional lead pencils is their affordability, but the downside is how frequently they need sharpening, and their impact on the environment. An alternative which I personally prefer is a good 2mm mechanical pencil (aka clutch pencil or leadholder), such as the one pictured below, with a special sharpener and boxes of leads. You can have just one such pencil and interchange leads as needed. Avoid those with thinner leads, such as 0.5 mm, because they can't be sharpened to a real point (0.5 mm is pretty blunt for our purposes!) and because you won't have a choice of hardness or softness.

Mechanical pencil

Straight Edge or Ruler

Strictly speaking, measurements are never used in geometry as they are not as accurate as proper constructions, and we are never going to use them in this course. We would have to go out of our way, though, to find a straight edge without measurement markings, so we might as well pick a good ruler.

For precision tools, you can't go wrong with brands that cater to architects, and every art shop will have at least one of those. You might wonder, wouldn't any ruler be good enough? Well, no: markings may not matter so much, but straightness is very important!

Here is how to test a ruler's straight edge: draw a line along the edge of the ruler, then turn the ruler around and draw a line on top of the first one along the same edge. I have tested this below with a trusty ruler I have been using since 1997:

Good ruler

Let's have a close-up look: see how the line is still definitely a single line? This means the edge is perfectly straight.

Good ruler close-up

Next I tested it with a metal edge, and you'll see why such edges are okay for cutting but should never be used for precision work.

Bad ruler

In the close-up above, see how the line splits towards the right? If the whole image could fit on this screen, you would see that even though the two lines are drawn along the same edge, they enclose a narrow space, indicating that the edge is slightly curved. Just what we want to avoid!

Compass

Our most interesting and most important tool is also the most costly one, but a good compass is worth its weight in gold and will last a lifetime. It's always fine to use a cheaper school-type compass for learning, of course, and upgrade when moving on to serious work (or when you get too frustrated with the lack of precision).

A compass basically has two legs connected by a hinge: one leg ends in a needle point, the other in a pencil point. The pencil leg can be adjusted to obtain different openings, and turned while the needle point is held steady on the paper, which creates a circle. It is entirely possible to create complex geometric figures with nothing but a compass, replicating the rope-and-pegs methods used in architecture a long time ago.

Compass and accessories

What to Look for in a Compass

  1. A screw mechanism to change the compass opening (or at least a screw to tighten the hinge so that it stays put once you've set the desired opening). You do not want a compass that opens and closes easily, because the opening will inevitably change while you work.
  2. An interchangeable pencil end. The end of the right arm of the compass above can be taken off and replaced with the little gadget on the right, in which any drawing tool can be inserted: pencil, pen, ruling pen, or even brush. This is incredibly useful, as the alternative would be to ink or color the lines freehand, endangering the perfection of the curve.
  3. An extension arm: This is the longer accessory at the bottom. It makes it possible to draw much larger circles. For instance, this compass can manage a circle with a radius of around 25 cm, but the extension arm stretches this to 35 cm.

Some compasses don't have a lead like this one does, but are designed to be fitted with a pencil. That's fine: it is then a matter of personal preference, subject to the same pros as cons as I have explained when comparing traditional pencils to mechanical pencils.

Tips for Using a Compass

  • Cover your work surface with a large piece of card or mount board (at least the size of your paper), both to protect it from the needle point, and also so that the point can penetrate enough to stay in its place. Otherwise it can get very frustrating, as it'll keep slipping out.
  • Place the needle or dry point, with great precision, where you want it to be, and then hold the handle (at the top) between thumb and forefinger to rotate it and create a circle. Getting a nice, even circle this way may take some practice at first—that's normal. Try to keep the compass reasonably upright while you draw. Never hold the compass with one leg in each hand, as that alters the opening.
  • I must stress this: take great care to place the needle point accurately, and to keep the pencil end sharp. The reason some people are good at geometry and some aren't is all down to precision.

Basic Constructions

That's enough theory, let's start drawing! Gather your tools and some cartridge paper, and let's get started.

Diagrams Legend

In the construction diagrams throughout the course, I use the following types and colours of lines. Here is what they mean:

Diagrams legend

Triangle (on a Given Side)

This is how to proceed if you're starting from a line segment, which means you already have one of the sides of the triangle.

Step 1

Dry point on A, draw an arc from B.

Triangle on a given side step 1

Step 2

Dry point on B, draw an arc from A to find the third point C.

Triangle on a given side step 2

Step 3

Join. If your compass opening is larger or smaller than AB, the triangle is isosceles.

Triangle on a given side step 3

Triangle (in a Circle)

If you have a given circle and you need to inscribe an equilateral triangle in it (meaning its three points will be on the circle), follow these steps:

Step 1

Draw a line through the center, cutting the circle at A and B.

Triangle in circle step 1

Step 2

With the same compass opening, draw an arc that cuts the circle at points C and D.

Triangle in circle step 2

Step 3

Join BCD.

Triangle in circle step 3

Perpendicular Bisector

This technical-sounding term refers to a line that does two things: it divides a segment (or an angle) in two equal lengths (or angles), and it is at a right angle to the segment it divides. This is a rather important device and is frequently used in the process of constructing other figures.

Step 1

With the point on A and the compass opening equal to AB, draw an arc.

Perpendicular bisector step 1

Step 2

Repeat with the point on B. The two arcs intersect above and below.

Perpendicular bisector step 2

Step 3

Join the two intersection points. The segment is now bisected and O is the mid-point between A and B.

Perpendicular bisector step 3

Tangent Through a Point on a Circle

If you have a given point (P) on a circle and need to draw the tangent through this particular point:

Step 1

Start by drawing the diameter that passes through P and the center O, and cuts the circle at another point A.

Tangent through a point on a circle step 1

Step 2

Set your compass opening to the distance AP, place the point on O and draw a large arc, almost a semicircle. It cuts the line AP at B.

Tangent through a point on a circle step 2

Step 3

Without changing the compass opening, place the point on B and cut the arc at C and D.

Tangent through a point on a circle step 3

Step 4

The line CD is your tangent at P.

Tangent through a point on a circle step 4

Tangent to a Circle From an Outside Point

Suppose now that P is a point outside the circle and you need to draw the tangent that passes through it:

Step 1

Join the segment PO.

Tangent to a circle from an outside point step 1

Step 2

Bisect PO at point A.

Tangent to a circle from an outside point step 2

Step 3

With the dry point on A and the opening set to AO, cut the circle at points B and C.

Tangent to a circle from an outside point step 3

Step 4

PB and PC are the two possible tangents from point P.

Tangent to a circle from an outside point step 4

Parallel (Through a Given Point)

Parallel lines are lines that never touch, so they travel in exactly the same direction. If your schooling was anything like mine, you were taught a vague shortcut to drawing them, but always ended up just relying on the grid printed inside your copybook. This, however, is the right and proper way of getting true parallels!

Let us start with a given line and suppose we have an outside point P through which the parallel needs to pass.

Step 1

With P as the center, draw any arc to cut the line at A.

Parallel through a given point step 1

Step 2

With the same compass opening, put the point on A and mark point B.

Parallel through a given point step 2

Step 3

Now place the point on B to draw an arc that passes through A and cuts the first arc at C.

Parallel through a given point step 3

Step 4

The line PC is your parallel.

Parallel through a given point step 4

Parallels (at a Given Distance)

Slightly more tricky is to draw a parallel at a specific distance from the original line.

Step 1

Start by marking two pairs of points on the line. Distances are not specific, but the further the pairs are from each other, the more accurate the result.

Parallels at a given distance step 1

Step 2

Find the bisector for each pair of points.

Parallels at a given distance step 2

Step 3

Open your compass to the desired distance and mark that distance on each of the two bisectors.

Parallels at a given distance step 3

Step 4

Join.

Parallels at a given distance step 4

Dividing a Segment

We'll finish this first lesson with a very nifty method to divide a segment into a number of equal parts. This is useful of course if you don't have a ruler with markings at hand, but even a ruler is no help if you have a segment measuring 5.63 cm which you need to divide into seven sections. This method is completely accurate and will spare you awkward calculations.

In the following example, we want to cut a segment AB into seven.

Dividing a segment step 1

Step 1

Draw two arcs with the point on A and B respectively. Their radius doesn't matter as long as they intersect.

Dividing a segment step 2

Step 2

Join A with one of the intersections and B with the other. This results in two parallel lines.

Dividing a segment step 3

Step 3

What we're going to do now is mark evenly-spaced points on each parallel, using the compass. The opening doesn't matter but keep it small so all the points fit on the line. Their number is [number of segment portions minus 1], which in the case of our example, is 7–1 = 6 points. Here the first point is marked from A.

Dividing a segment step 4

Step 4

Move the compass point to the point just marked, and mark another, then repeat till six points are marked, then do the same starting from B.

Dividing a segment step 5

Step 5

Connect the points, and the lines cut the segment into seven equal parts.

Dividing a segment step 6

So, we have taken our first steps into geometry as an art, with basic operations that will come in handy in future lessons or in your own explorations. Next time we will be jumping right into actual shapes and patterns, working with the numbers 4 and 8...

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