Bird (paper crane): 4 branches (corner branch), 1 short branch (center branch).
Fish: 4 angular branches, two long and two short.
Frog: 4 corner branches, 1 central branch, equal length. (Four short side branches can be organized around it)
Double square, mine (double triangle).
The right way is:
Learn mathematics, physics and chemistry well, and you will not be afraid of traveling around the world.
If you practice the basic form well, Xipi can do any variation.
2. Splitting and grafting 1. Point splitting.
Usually occurs at the apex of a branch. Increase the number of short branches by sacrificing branch length.
The pictures shown here are only examples of the most perfect point splitting. (The branch length ratio is the largest after division)
Some CP pictures can clearly see the effect of splitting, and they add countless details:
This cp splits various branches in various ways, and even splits layer by layer on the right. Comparing it with the one on the left, you will find that it is actually a basic bird type, but it keeps splitting and changing into so many branches and details.
The structure of splitting changes in various ways, and it is impossible to explain them one by one. 2. Grafting
While not changing the original structure, graft other structures to express details. It can be divided into external grafting and internal grafting.
You can graft branches with very large proportions to form the whole, and let the most basic structure only determine the framework of the model, accounting for the smallest proportion.
The picture above shows the process of grafting bird claws on a basic paper crane. The right side is the outline of the paper CP after the grafting is completed.
But Lao Luo said, the optimal solution to origami design is the solution that maximizes paper utilization. Therefore, Lao Luo would definitely not look good with the extra layer of paper grafted. So he used the extra layers of paper to create details for the wings, tail and head.
Another example to explain grafting is the example of a lizard grafting its toes. Let’s take a look at his basic type first:
It is not difficult to see that the densest places in cp are the endpoints of branches. .
After simplification, the three surrounding corners are actually useless and do not need to be considered. All that is used for the main branch distribution is a pentagonal box against the lower left. That is the figure enclosed by the highlighted lines in the figure below.
The picture above shows the grafting of a two-toed structure. Such grafting is called external grafting. The paper is extended from the outside of the square to complete the structural grafting.
Obviously, the efficiency of such grafting is extremely low. Because as you can see, the basic lizard model already has no shortage of other details. Enlarging the paper is useless and will only reduce the utilization rate of paper.
So Lao Luo said, we can still graft like this:
Witty! Perfect! Lao Luo said, you first cut along the axis and cut into multiple tiles. Then graft the rows that need to be grafted. As shown above.
But Lao Luo said that it was too difficult to fold it like this. Can't we choose a simple method of taking the line?
So, he cut horizontally and vertically and inserted the structure. This kind of grafting is obviously more pleasing to the eye and more feasible.
However, Lao Luo showed an evil smile. You guys! Too Naive!
Use your brain, you won’t cut it like this!
This is a perfect graft that combines utilization and feasibility.
I will throw a baby CP to you before the end. (I forgot to post it in the previous section!!!)
Those with a discerning eye can see that this is actually a CP modified from a basic eccentric mine with a bellows structure.
It corresponds to the old saying:
Practice the basic form well. blablabla. .
Lao Luohe’s grafting structure.
Its simply abnormal.
The highlighted area is the basic shape of the bird. The basic basic structure only plays a structural guiding role in this work.
All details are completed by grafting other structures, including the extension of the neck (center branch of the square), wing feathers (left and right above the square), bird claws (square left and right below).
Insert a piece of knowledge about tree diagrams.
***Dendogram
A dendrogram consists of a trunk, branches, and branches. (Picture)
In order to facilitate memory, it can be specified as follows:
Leaf node: vertex (leaf point) - the terminal vertex of all branches
Leaf edge : Branch (leaf edge) - indicates the part of the branch.
Branch node: bifurcation point (branch point) - indicates the point where a branch meets a branch or a river.
Branch edge: River (branch edge) - represents the distance between branches, that is, the river.
Weight: Length (right) - the term "right" is too technical. . Just remember this is the length. .
There are many remaining mathematical theories, including weird content, so you can read it by yourself. .
When drawing a tree diagram, pay attention to drawing all branches separately, even branches that are wrapped around each other. (If not in use, you can hide it by folding it.)
Treemap is the most convenient and easy-to-use basic analysis tool. In CP creation, sometimes you have to design through tree diagrams. (For example, designing snake belly text)
The process of tiling the tree map.
The process of converting a tree diagram into a circle diagram.
Theoretically, all tree diagrams can be obtained by folding a square. Theoretically, the basic form of a single tree diagram can be obtained from multiple cp diagrams.
Chapter 3 Axial Polygons, the axes and ridges mentioned, although the concepts are important, there is no need to memorize these properties in practical applications. pass.
To put it roughly, when you fold the work, you will feel that a basic shape is symmetrical along a certain line. That line has the most paper layers along it, and it is a bit difficult to fold it in half. That line is called the axis. . A single-axis polygon has only one axis or multiple axes. (Represented by the green line in the book)
The basic outline outside the axis is called the ridge. (Indicated by the red line in the book)
4. Circle River Packing Method (Circle Packing Method)
Get to the point.
The source of this method:
Assume that the length of a corner branch is L. If a corner of a square Fold in half an infinite number of times, then the edge of the branch on the cp diagram is close to a quarter of a circle with a radius of L.
This theory holds true for both edge branches and central branches:
With such a theoretical basis, it is not difficult for us to find:
If I need a basic type with n branches of length a1, a2, a3...an, I only need to arrange n branches with radius a1, a2, a3...an in a square Circle, you can create and adjust CP.
All circles meet the following characteristics:
1. The center of the circle remains within the square, or the edges and on the vertex.
2. Circles cannot overlap. (Overlapping will affect the structure and branch length)
3. The cp line that crosses the circle always passes through the center of the circle.
The rest is the river. The representation of a river in the circle river method is a curve with a width. Its width is the length of the river in the tree diagram.
Rivers satisfy the following characteristics:
1. All rivers flow from one side of the paper to the other. Split the paper into two parts.
(In special circumstances, the rivers are connected end to end, but can still split the paper into two independent parts.)
2. The cp line crossing a river is always perpendicular to both sides of the river.
After talking about the tree diagram, let me insert a gesture:
About the grafting of fold structures
The picture above shows two common fold structures.
The first picture is the structure of scales, which can be found in cobras, dragon gods, koi carps, morning dragon scales, ornate conches, etc. (The book talks about the method of processing fish scales, which is very detailed)
The second picture is the folds of tortoise shell.
Other common fold combinations can also be extremely gorgeous, and are generally called flat inlaid origami by our enthusiasts.
Uncommon combination, masters can use it to graft facial features on a big face, details on clothes, and other things.
5. Molecular line chart
It is used to bring the starting points of all branches excluding rivers to one point. For example, in the following example, the four circles in the basic mine type are tangent to each other, but not to each other. In this case, a molecular polyline diagram is used for transformation.
(Of course, you can also fill in the gaps by filling in the circle and river. This process is called the stump method. It is mentioned in Chapter 11.)
Take the special basic type-mine molecules as an example:
General symmetrical graphics are used as the basic type of mines. No matter how you want to arrange the branches, the aggregated branches will still be symmetrical lengths.
Therefore, you need a structure to distinguish the branch lengths and fill the error caused by the gap in the middle of the circle river.
As can be seen from the above figure, the determination of the structure of the molecular polyline depends on the tangent between the circle and the circle.
The following are variants of this structure. The branch lengths they obtained did not change, but the morphology was different.
The above molecular fold line that occurs in the middle is called a gusseted molecule.
The above structure is called an arrow molecule. This structure can be cut into two parts: one part is kite-shaped, and the other part is grafted onto the arrow-shaped part. Therefore, the molecular structure is also divided into pure type and mixed type.
The following is a pentagonal molecular line chart structure.
But after talking so much, you are still confused.
So remember the wiring steps of the molecular line chart above. When you encounter similar problems in your design and need to adjust the branch length or fill in the blanks, consider the molecular structure. (Or use the stump method mentioned and add circle fill.)
This ends the knowledge about traditional basic types and cp. Therefore, let me sort it out for everyone -
The steps for designing general traditional cp:
1. Draw branches The diagram, that is, the tree diagram, clearly identifies the branch lengths (the smaller the length value, the better. For example, if the length is 1 unit, it is easy to arrange circles).
2. Think about and build the relationship between circles and rivers in squares. Arrange. ① Try to arrange large branches on the corners and edges of the square. ② Then arrange branches of medium length and insert small branches indirectly. ③ Pay attention to the basic branch arrangement. For example, do not draw the branch on the left hand of the character on the right side of the body. ( This step requires brain cells and time. If necessary, you need to adjust the length of the branches of the tree diagram.)
3. Connect the centers of tangent circles (including those that are tangent across rivers). Let the squares split to form countless tiles .(These lines will become the axes)
4. Conclude the angle bisectors of all the corners of the tile.
5. Conclude all the tangents of the tangent circles.
6. Combine steps 4 and 5 The resulting line is transformed into a molecular polyline diagram and fills in the excess paper layers of non-tangent circles.
7. Draw the basic cp and consider point splitting or grafting structures.
However. This involves circles, which are related to The calculation of circles has always been a troublesome part for people. Therefore, the cp drawn by TreeMaker software is more of the legendary data flow cp: it is difficult to pick up points.
So I believe you will pass it by just taking a look.
But don’t worry. We also have a simplified version of the original method:
1. Draw a branch diagram and determine the basic structure.
2. Select the appropriate basic type as the structure among the traditional basic types.
3. Consider grafting other basic types or structures, and consider splitting.
When you become proficient, in fact, of the two methods, the second is simpler. The old Luo He mentioned earlier is very It is obviously original through the second method, and the effect is still great, great, and great.
Just like what Lao Luo said about the snake belly -
Sometimes, it is worth sacrificing some efficiency in exchange for something else. .
The operability and convenience make cp easier. So what exactly is the annoying little goblin Snake Belly?
6. Snake Belly Box Pleating
Snake Belly only has simplicity Geometric lines and diagonals (common quadrilateral accordions are horizontal, vertical and diagonal lines).
The work that first introduced the accordion theory to people’s attention was the Moser train. Although he used a rectangle.
(Mosser woke up laughing from his dream: I just picked up a train and seemed to have discovered a new world!)
Wow, this announcement is shocking Everyone jumped. This box structure actually has endless possibilities.
*Lao Luo’s book mentions an ancient box, also known as the snake belly structure, but it was published much earlier than this theory. By studying the box, Lao Luo discovered the possibility of the snakes belly.
Advantages of snake belly:
As long as the number of squares is sufficient, you can graft rivers and change branch lengths at will, without excessive arrangements and calculations. The structure is incredibly flexible.
If you like, you can graft an infinite number of origami cranes onto one piece of paper. Because the basic lines of the snake belly are horizontal and vertical, there is no need to worry about increasing the length or grafting the structure, which will cause confusion to the original structure. Simple and very convenient.
It is precisely because of this that the snake belly structure has endless possibilities and variability, and is widely loved by modern origami people.
Lao Luo called the snake belly an epoch-making theoretical discovery.
Disadvantages of snake belly:
Sacrifice the utilization of paper.
Branches stick to regular shapes. (Can be modified with special structures) The circle river method in the belly of the snake is transformed into the polygonal method. (The quadrilateral bellows is represented by a circle turning into a square, and the river curve turning into a polyline)
The thickness of the river in the bellows is a fold line parallel to the side in the river, and the number of grids it passes through x the thickness of the paper. So the longer the river, the greater the thickness.
The following example is a simple example of snake belly design. Once you learn it, you can design characters!
1. Draw a tree diagram. (The branch length is half the side length of the square)
Second, consider distribution. Pay attention to the relative positions of left, right and up and down, and ensure that the center of the square is within the paper.
Third, connect the center of the square to the vertex of the square, and distribute the diagonal lines according to the refraction law of diagonal lines.
(Actually, people with eyesight can see this. For the square that is close to the edge of the paper, the diagonal line does not need to be extended to the edge of the paper, it can be extended to the center. The one on the lower right of the picture above For the largest square, the two rightmost diagonal lines can be discarded.) In the polygon design rules, when designing with cp, the distribution of diagonal lines is reflected in: reflection when meeting other branching squares and rivers. (Of course, I added this myself, and it is not mentioned in the book.) The intersection of the two oblique lines must be the vertex of the branch square, or a certain inflection point of the river fold line.
A simpler way to draw is - branch square: connect the center of the square and the four vertices. River polyline: connects corresponding vertices.
The refraction law of oblique lines:
The oblique lines in the snakes belly are distributed according to the refraction law.
When a straight line meets a diagonal line in the process, it travels in the direction of refraction. When two diagonal lines intersect, there will be no more.
When a straight line meets the intersection of three lines, it is like the example of opening direction and straight direction traveling.
Of course, it is also possible to encounter four fold lines that intersect. At this time, they extend in the direction of the other three openings. This is the common central branch of the snake belly (also a branch structure that gives many beginners a headache). If you just want to design a simple snake belly piece, then the above theory is enough for you.
If you say, I am not satisfied, I want something more exciting.
Come, I will satisfy you.
Level switching in the belly of the snake: This structure is used to deal with the problem of missing width. (That is, what is often called local high equal division, local low equal division, and more other special structures.)
The above picture describes the simplest level switching. The number marked on the edge indicates the thickness (height) at that location. Readers can try by themselves according to the solid line valley line and the dotted peak line aggregation. (The green and gray lines are auxiliary lines)
The following figure introduces the two most common structures that can be used to change the hierarchy. btw, Dragon Gods scales also have level switching, but that is a more advanced transformation method.
I'm not good at this stuff, so I won't talk much about it. You can look at other cp examples to find the hierarchical switching structure. Tight winding path:
The tight bends of the river are formed bystructure. Can be used for special structures, or to fill redundant gaps in CP.
The most well-known representative act: Angel Wings.
Of course, this does not mean that the labyrinth structure can only be used here.
Lao Luo gave such a design in his book, and the cp was taken from Lao Luo’s cicada nymph. (It’s super delicious when fried!)
As you can see in the picture above, Lao Luo is trying to use a winding structure to design the abdominal texture of the cicada nymph, which requires many small branches. After the CP was drawn, Lao Luo found that many places did not match the original CP because the abdominal line was twice the original bisection and was exactly half a grid off the original CP line, making it impossible to graft.
But the great gods are all awesome. Lao Luo came up with the following solution. (It is also a hierarchical switching structure)
The cp on the right adds a triangle to perfectly integrate the staggered lines into the original cp.. Doing this does not affect the length of the small branch, because the square overlap we talked about before, the river in cp does not overlap with the circle of the branch. The place on the left where no structure is added is offset by half a grid as a comparison. Polygonal accordion:
(You will never want to touch the hexagonal accordion, so I will conclude.)
The hexagonal accordion leads to the fact that you can deflect the accordion CP structure in any way angle to achieve different effects.
In the picture above, after the deflection angle, the thorns on the legs are naturally arranged separately.
The same structure can also be found in Tetsushi Kamiyas Phoenix Wings. Including Origami Bar Red God’s original phoenix, various complex bugs, etc. Overlapping Squares:
Although the circle of the circle rule becomes a square, in essence, the branch length is still the radius of the circle. Therefore, the squares can overlap while ensuring that the branch circles do not overlap. (Among them, circles become perfect overlaps when they are tangent)
Perfect stretch here.
This structure is called Pythagorean stretching (What the hell!)
Determining the overlapping cp also involves some complex calculations. I believe you won’t read it, and I won’t write it either. .
I haven’t done it before, so I can’t say what the problem might be.
So it is best to design an overlapping structure, and then do a local CP to test whether your overlapping structure is feasible, and then apply it to the overall CP.
Example:
The last two pictures are a mixture of fish shape and kite shape. The content of the book roughly ends here.
I believe that readers who have gained some insights after reading this book will already have the ability to create some simple things.
Original design is not terrible, the important thing is to dare to try.
Draw some simple structures, do some simple CP, graft some simple structures, even if you don’t graft any structures, just connect a circle of paper on the outside to turn the two colors.
In fact, many designs are grafted with a layer of paper on the outer layer to create two colors. They twisted the original CP at an angle and put it into a larger square.
In fact, it has been developing very rapidly since its belly was folded. But I have never seen a theoretical compilation like this book to tell us more clearly how the theories of snake belly and origami have changed over the years.
Perhaps there may be no qualitative innovation. But origami doesn’t stop making progress.
Because the possibilities with paper are endless.
