Content from Robert J Lang The Secrets of Origami Design. Almost all the pictures are screenshots from the book. Let me state in advance that this article is an extremely concise introduction to part of the structure and original content, mainly including practical content such as basic CP models, design tools, and design methods. (It is probably content that the great masters despise, but that newcomers cannot access, as content that must be understood to advance to the great masters.) Most of the concepts involved in the original book are not mentioned, and because there is no domestic terminology translation standard, this post All terms in the book are translated for easy understanding and understanding. If you have any questions about the content in the book, please refer to the original English book.
1. Basic type
Traditional basic types include kite, bird, fish and frog basic types. Simpler basic types include the cabinet (half of the kite type), the double square, the mine basic type, and the uncommon flat basic type (a square basic type obtained by folding the four vertices of a square multiple times like the center).
Kite: flat type.
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 all over the world.
If you practice the basic form well, Xipi can change it in any way.
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 division, and he added countless details:
This cp splits various branches in various ways, and even splits layer by layer on the right side. 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 cannot be explained 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 a 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 of grafting is the grafting of a lizards toe. 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:
Wise! 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, it is too difficult to fold 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 and you won’t do it like this!
This is a perfect graft that combines utilization and feasibility.
Toss you a baby cp 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 the basic type of eccentric mine with a bellows structure.
As the old saying goes:
Practice the basic forms well. blablabla. .
Lao Luohe’s grafting structure.
Simply perverted.
The highlighted area is the basic bird type. 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), and bird claws (left and right below the square).
Insert a piece of knowledge about tree diagrams.
***Treemap
A dendrogram consists of a trunk, branches, and branches. (Picture)
In order to facilitate memory, the following provisions can be made:
Leaf node: vertex (leaf point) - the terminal vertex of all branches
Leaf edge: Branch (leaf edge) - represents the part of the branch.
Branch node: bifurcation point (branch point)-indicates the point where a branch connects to 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 other mathematical theories and weird content left to read by yourself. .
When drawing a tree diagram, be sure to draw all branches separately, even those that wrap 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 treemap tiling.
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, it is not necessary to remember 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. (Indicated 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 an angular branch is L. If one corner of a square is folded in half infinite 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 the edge branch and the central branch:
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 circles with radius a1, a2, a3...an in the square, and thats it Created and adjusted CP.
All circles satisfy the following characteristics:
1. The center of the circle remains within the square, or on the edge and vertex.
2. Circles cannot overlap. (Overlap will affect structure and branch length)
3. The cp line passing through a 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 meet the following characteristics:
1. All rivers flow from one side of the paper to the other, splitting 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, insert a posture:
About grafting of fold structures
The picture above shows two common fold structures.
The first picture is the structure of scales, which are found in cobras, dragon gods, koi fish, morning dragon scales, ornate conches, etc. (The book talks about how to deal with fish scales, which is very detailed)
The second picture shows the folds of a tortoise shell.
Other common pleat combinations can also be extremely gorgeous and are generally loved by usThe best ones are called flat inlaid origami.
Uncommon combinations, masters can use to graft facial features on a big face, details on clothes, and other things.
5. Molecular line chart
Used to bring the starting points of all branches without rivers together 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 circles and rivers. 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 tree stump method mentioned and add circle filling.)
This ends the knowledge about traditional basic types and cp. Therefore, let me sort it out for everyone——
Steps for designing general traditional CP:
1. Draw a branch diagram, that is, a tree diagram, and specify the branch lengths (the smaller the length value, the better. For example, if the length is 1 unit, it will be easy to arrange circles).
2. Think about and build the arrangement of circles and rivers in the square. ① Try to arrange large branches on the corners and edges of the square. ② Then arrange the 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 character’s left hand on the right side of the body. (This step requires brain cells and time, and the length of the tree branch needs to be adjusted if necessary.)
3. Connect the centers of tangent circles (including tangent circles across rivers). Let the square split to form an infinite number of tiles. (These lines will become the axes)
4. Draw the angle bisectors of all corners of the tile.
5. Draw all tangent lines of the tangent circle.
6. Transform the lines obtained in steps 4 and 5 into a molecular polyline diagram and fill in the excess paper layers that are not tangent circles.
7. Draw the basic cp and consider point splitting or grafting structures.
But. This involves circles, about circlesThe calculation has always been a headache for people. Therefore, the cp drawn by TreeMaker software is more of the legendary data flow cp: it is difficult to pick out the points.
So I believe you just have a look and pass by.
But don't rush. 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 point splitting.
When you become proficient, the second method is actually easier. The Lao Luo He mentioned earlier was obviously created through the second method, and the effect is still great, great, great.
Just like Lao Luo’s evaluation of snake belly——
Sometimes its worth sacrificing some efficiency for something else.
The operability and convenience make cp easier.
So what exactly is this annoying little goblin in Snake Belly?
6. Box Pleating
The belly of the snake has only simple geometric lines and diagonal lines (the common quadrilateral snake belly is horizontal, vertical and diagonal lines).
The work that first introduced the snake belly theory to peoples attention was the Moser Train. Although he used a rectangle.
(Mosser woke up from his dream with a smile: I just picked up a train and seemed to have discovered a new world!)
Wow, this announcement shocked everyone. 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 paper 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 paper utilization.
Branches adhere to regular shapes. (Can be modified with special structures) The circle river law in the snakes belly is transformed into a polygon law. (The quadrilateral bellows becomes a circle and becomes a square, and the river curve becomes a polyline)
The thickness of the river in the snakes belly is determined by taking 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 a 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 and right and up and down, and make sure 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 oblique lines according to the refraction law of oblique lines.
(Actually, people with a discerning eye can see this. For the square that is close to the edge of the paper, the diagonal lines do not need to be extended to the edge of the paper. They can be extended to the center. The largest square in the lower right corner of the picture above has two diagonal lines on the far right. It can be discarded.) In the polygon design rules, when designing CP, the distribution of diagonal lines is reflected in: reflection when meeting other branch 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 law of refraction of oblique lines:
The oblique lines in the snakes belly are distributed according to the law of refraction.
When a straight line encounters 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 an 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 on, I'll satisfy you.
Level switching in the belly of the snake: This structure is used to deal with the problem of missing width. (That is, the commonly referred to local high equal parts, local low equal parts, and more other special structures.structure. )
The above figure depicts 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 figure below illustrates the two most common structures that can be used to change levels. btw, Dragon Gods scales also have level switching, but that is a more advanced transformation method.
I’m not good at this stuff either, so I won’t talk much about it. You can look at other cp examples to find the hierarchical switching structure. Compact labyrinth:
The tight bends of the river form a structure. Can be used for special structures, or to fill redundant gaps in CP.
The most well-known representative act: Angel Wings.
WhenHowever, this does not mean that the maze structure can only be used here.
Lao Luo gave such a design in his book, and the cp was taken from Lao Luos 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 masters are all awesome. Lao Luo came up with the following solution. (It is also a hierarchical switching structure)
By adding a triangle to the cp on the right, the staggered lines can be perfectly integrated 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 definitely don’t want to touch the hexagonal snake belly, so I will conclude.)
What comes out of the hexagonal bellows is that you can deflect the bellows CP structure to any 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 Tetsu 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 will not read it, and I will not write it down.
I haven't done it before, so I can't say what the problem will 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 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, and graft some simple structures. Even if you don’t graft any structures, you just connect a circle of paper to 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 then 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.
There may be no qualitative innovation. But origami doesn’t stop making progress.
Because the possibilities with paper are endless.
