God Save the Square

My good friend Padyta asked me if it was possible to obtain a rectangular angle from an arbitrary piece of paper. If we remember that a square is possible to be constructed from that angle, her question have a great importance.

To build a perpendicular line to a given one in two-dimension geometry demands the use of ruler and compass and it is not an easy thing. One way is tracing a circumference of an arbitrary radius, centered in an starting point A, then a second circumference centered on the intersection of the line and the first circumference (point B), the intersection of both circumferences gives us a point C, equidistant to both corners A and B, a new circumference of the same radius centered on C and its intersection with the first circumference gives us a point D and finally the intersection of a circumference centered on D with the one centered on C gives us the point E, perpendicular to the line AB.



However, the "origeometry" allows us to use the paper itself and gives us a third dimension and increas our capabilities. So, if we have a given straight line, folding the paper aligning the line on itself we can get a perpendicular line.





Now that we have our rectangular angle, is it possible to construct a perfect square from it? The answer, luckily, it's a yes. Here it comes a way.

First you need to fold a 45° line between both lines, that will allow us to find the main diagonal passing through the next corner of the square. To do it, we fold aligning both lines on their intersecton point.



Then we align the small diagonal on itself folding a diagonal that passes over the corner of the square. A good choice of that corner will allow us to optimize the use of the paper to get the biggest size possible.





We can cut the main angle and the sides of the square. As we saw on the previous post we can obtain the opposite angle and complete the square by folding the main diagonal aligning it on itself, passing through the original corner.





It's pendant to btain a method to optimize the size of the obtained square, maximizing its area on the given paper. Many regards.

Our friend the Square

Since traditional Origami works over a perfectly squared piece of paper, the issue of obtaining it has become one of the greatest importance. How many of us has coursed to the heavens when we realize that our "special for origami" sheet is nor even a rectangle but a totally irregullar quadrilateral figure?

Through the years I've used diferent methods to "square" this sheets, with varied results; for example if the sheet is a regullar rectangle you can fold the bisector of one of the corners, this is one of the main diagonals, the intersection of it with the opposite side will determine the other corner of the square.




There is also the case of a long sheet came from a roll of paper, where you know that two opposite sides are perfectly parallel, then it's only necessary to fold a median perpendicular to both edges alining them, and then cutting both layers in a distance equals to half the height of the sheet, as is showed below; of course you will need a good graduated rule for this.



However, these methods fail when our sheet is an irregullar quadrilateral, like the fellow in the heading of this post.

To find a safe way to rescue the hidden square of our paper I went back to the teachings of my old Math's school professor (Carlos "the lizard" Zuñiga): "to construct a geometrical figure go back to the characteristic that defines it, the one that give it the being..." in this case, the 90° angles of its corners. The idea then is to inscribe a rectangular angle on one of their corners with a set square (or the corner of a copy machine sheet of paper)




we cut this angle. Over it you can fold the main diagonal, aligning one side on the other, this diagonal will be our future simmetry axis.




Then, to construct the second diagonal, we reflects the rectangular angle on the opposite side by folding in the closer adyacent corner and aligning the simmetry axis over itself, obtaining in that way a perfectly inscribed square to be cutted.




Finally to remark that if you use one of the side to inscribe the first rectangular angle you save one of the cuttings.


I hope it works for you all ;) many regards

The Paper (Chapter 4)

It's been a while, tons of things, but here we are. After a well deserved rest and my journey to the spectacular Cusco Peru Convention (which I will write a couple of things as soon as I get the pictures) it's time to continue this road. We left it talking about some of the structural properties of the paper, grammage, formation and thickness, and yet it's necessary to mention others that complete the structural framework of this noble material.

So, we will take care now of Directionality, Double Face, Smoothness and Porosity, all affected in a direct way by the productive processes in paper making.

Directionality

This is one of the main characteristics on papermaking. Due to the fact that pulp moves in a fitted direction in shaking, beating and production lines, the fibres get aligned in a Main Direction or Grain Direction (the flux one), affecting most of the mechanical properties of paper, such as Tensile Strength, fold elasticity, form and propagation of tears and many others. Also folding, marking and punching are done easily in this axial direction. It is so important for box making, packing and forming industries that it is indicated in cardboard packages; in case of rolls this direction is obvious.

To determine the Grain Direction the easiest way is to observe the natural bending of the sheet in different directions, holding it in one of its edges, here an example with a square of bond printing sheet

the Grain Direction offers a resistance to bending greater than the Cross Direction.

Another way to know it is observing the curling of paper when hydrated, fibres expands greater in its axial direction than the cross one, producing an abnormal and oriented deformation of the sheet, as you can see in these old books victims of humidity:

Finally, other tests to obtain it are more destructive and includes the measurement of tensile strnght (greater in the MD) or the rupture line of cracks when bursting (paradoxically perpendicular to Grain Direction)

We will talk a little more of this when we faces the mechanical properties in next entries but, as an example, if we get a kraft sheet of of 90 (g/m2) paper that, as we saw before, is a chemically processed paper, which preserves the fibres almost undamaged, we could see that its tensile strength is 7.43 (kN/m) in MD and 3.81 (kN/m) in its Cross Direction. Similar differences can be seen in many other properties, as foldability, tear propagation and others we will see.

Double Face

Also due to papermaking processes, the pulp is moved in its line of production through bands with metallic grids, which affects the concentration of elements in one surface and the opposite, more in contact with poles and rolls. It's possible to show a distribution of components along the z-axis (or thickness axis) of the sheet, been the main affected fine and fillers.


The Grid side is called the Wire Side and the one in contact with poles the Felt Side. The first graph shows the ash concentration, which is normally used as a filler, and the second one the Resistance to filtration, greater in the presence of fine, meaning that paper absorbs less in the Felt Side than in the Wire Side, affecting inking and coating.

To identify one side from the other is important to know that there are surface differences, the grid leaves marks (squares or diamonds), sometimes visible at counterlight, you can use light carbon pencils or markers also. In adittion, Felt Side normally shows a closer formation and greater softness and smoothness than Wire Side.

Another method is to make a tear holding one side against a glass or a rigid surface, first in the MD and then changing to the CD (for example in a corner of the paper); when the Wire Side is up, the tear (specially on the curve area) shows a greater feathering; finally, if paper contains an abrasive filler, marking a line with a silver coin will live a darker mark on the Felt Side.

Double Face affects directly in the best suited type of fold and their resistance, Felt Side is denser and then it resist more the valley folds (compression) than the mountain (expansion), among other characteristics.

Smoothness

Smoothness is a surface property and in general has a greater influence in the mechanical use of paper, writing and manipulating than for folding and forming, however, aesthetic effects are important and also the presence of coating to improve surfaces affects on folding. There are several methods to measure smoothness, as optical analysis with light, the use of microscopes and micrometers, the pass of air trough paper and glass, friction against metal or other paper or inking with roll are also used.

Porosity

Porosity is a vital factor in paper composition, one must consider that paper is 50% air, some of it is inside fibres, but most of it is in the pores between them and its components. The rate between pore volume and total is called porosity. But it is only measured in laboratories, tha value normally measured is air permeability, which is basically the flux of air that passes at pressure trough a sheet of paper, or its inverse, the resistance to the pass of air. These parameters are related, but very different, paper of same porosity but with different size and distribution of pores can have very different air permeabilities.

Porosity is important for many reasons, as its effects on mechanical properties, like tensile strenght, or superficial, like appearance or printing, saturation or coating treatments, also for the usage of paper (tissues, towels, cigarettes, absorbent paper, filters or isolating paper).

It is important to note finally that foldability and tensile strength diminishes with porosity and also the number of possible foldings and handling of paper for complex origami models, along with the poor marking of folds due to the lack of fibres and material (try a komatsu on a tissue towel); however, a too low porosity, a dense paper, can make also very difficult its handling.

The Paper (part III)

It is curious how an investigation brings things up. As you are reading new sources and getting some info, a book falls into your hands refuting facts that you considered for sure, clear and concrete. This happened to me with “Properties of Paper: An Introduction”, published by TAPPI (Technology Association for Pulp and Paper Industry), from authors James C. Abott and Stanley Trosset; a fantastic book, even when it is more oriented to industrial papermaking on USA, leaving behind artistic uses and handmade paper.

So much new information, incomplete matters, that I've been forced to speak a little more about nature of paper and its making.

We already said that main source for cellulose is treewood. Also saw that together with cellulose other elements coexists in plant's cells, like Hemilcellulose and Lignin. However, reading this book I've learned that Lignin works as a binding agent between fibers and that its concentration grows in the outer layers of the cell, contrary to what I said in the previous entry. In fact, it is necessary to remove lignin to free the fibers in an homogeneous solution, since it concentrates on the outer walls of the fiber isolating it from others, reducing the number of possible contact points between them (which are what generates the "bond" characteristic of paper). Lignin has a very complex and variable chemical structure, it is insoluble in water but can be disolved on certain acids solutions.

As a tip to whom desires to cook their own pulp, an agent that disolves lignin (not very efficiently but it works) is popular caustic soda in water. Its efficiency improves in pressence of high temperatures and pressure (but to use a pressure cooker with caustic soda is extremely dangerous!). Do not worry about cellulose, its chemical binding is strong enough to all acids except a few number of strong ones.

The other component, Hemilcellulose, is a polymer similar to cellulose but with slightly different molecular arrangements; is important in paper since it stimulates creation of fibre to fibre contacts and in its water absorption hability, however, it doesn't resist the lignin elimination process.

Then, if you want to make housemade paper, forget about tree wood and look for a plant with low concentrations of Lignin and fibres not too longs (that leaves out cotton), because that will result on a too resistent paper, like cloth, hard to fold (don't forget that the mark of a fold is given by the fracture of the fibre).

Also because the fact that papermaking from tree wood has become an extremely contaminant process just for the use of chemicals in the removal of lignin, and it is in that way because the low costs induced by wood and the chemical usage. Among the ways to produce cellulose there is the pure mechanical (as stone groundwood and Refiner Mechanical Pulp), constituting 10-15% of world's paper production, their disadvantages are the difficulties to obtain an uniform and homogeneous pulp, the breaking and damaging of fibres and paper's short lifetime, reasons because it is used mainly for newspapers, catalogs and light publications.

Lignin gives paper and pulp a characteristic brown color. It's important to understand that is impossible to remove all lignin from pulp, so all pulps coming from tree wood have that characteristic before the bleaching process; kraft paper has that colour only to the absence of the bleaching stage and it is a pure chemical pulp process.

As we talk about pollution, the whole process of papermaking is extremely water and energy consuming and, even when enormous advances has been developed in the reusage of water and chemicals to reduce that fact, it is far away from being reasonable for nature., all that leads us to the need of minimize the use of paper in our regular life and to respect and love the piece of paper we hold in hands, because the cost and sacrifice that it meant.

Properties of Paper

At last I think we can enter the land of the physical and structural differences between the different types of paper. The Characteristics of paper are Structural and Mechanicals, in the next posts I will talk about each one of them.

Structural Properties are:

Grammage
Formation
Thickness
Directionality
Two-sidedness
Smoothness
Porosity

Grammage

In industrial terms, weight of paper is measured by fixed packages of a given number of sheets and standard sizes for them, which makes thorny and difficult to make any comparative analysis between different kinds of paper; for that reason, the International System of Measurements (SI) established the mass content in a single sheet of known area, or grammage (grams per square meter, g/m2) as the mass - characterizing value for paper.

Typical values of grammage for paper are:

Formation

Formation on paper refers to uniformity in the distribution of the fibres and other components along and wide the sheet of paper; it is in the pulp sheet making stage when this is controlled.

Thickness

Thickness (or Caliper) is a vital parameter but commonly undervalued by designers and folders, who normally looks for the thinness possible piece of paper to its much complex models, specially for the box-pleated ones. There is a tendency to believe that thickness of paper is negligible and it will not affect the geometry and symmetry of the figure, nothing further from reality, specially when folding lots of layers together.

It is obvious that the maximum number of layers to be folded together is related with the thickness of the paper. However, there is a physical limit that show us to understand the real importance of it when folding any sheet. To illustrate it let's exaggerate the thicknes and see how many layers creates long and wide concentric circumferences around the fold; the long the number of them, greater the amount of paper they take away from the model.

The calculations of these diameters were done by an american mathematics student (now teacher) called Britney Gallivan on Dec 2001, getting a numeric series and a formula to obtain the needed long L for folding a piece of paper of thickness t a consecutive number n of times:

For example, regular thickness of a bond printer paper is 0.1 mm, let's assume we want to fold it 10 times over itself, we will lose exactly 55.036036 meters! of paper only in the foldings...

She achieved a world record folding a sheet over itself a number of 12 times (for curiouses our bond paper should be 880 meters!!, nearly a kilometre!).

Regular values for thickness are given in the next table:

An important parameter results from diving grammage by the thickness of a paper, called apparent density, thinner papers but with high grammages can be more resistant to tearing; generally is in compression stage when this characteristic is achieved. We will see this when mechanical properties are reviewed.

The Paper (part II)

I've planned to speak a couple of things about the effects on paper of folding but then there are still one or two things still to say about paper itself and its making. Most of this comes from the fantastic Michael G. LaFosse's book "Advanced Origami" and his project Origamido.com, book which is an ultimate guide to become an artisan of papermaking and origami.

Cellulose

Continuing with our approach to paper's nature, in the first part we mentioned cellulose and its discovering in 1852 by Meillier; however, some sources points to the french biogolist Anselme Payen as the first person isolating cellulose from wood [1, 2]. This element constitutes the main component of plant's cells and is important to origami since its atomics bounds promotes the creation of long molecular fibers. Looking at the cellulose molecule (C6 H10 O5)



is possible to note that their free OH radicals can generate a very strong H2O bound and also an oriented chain, which is the origin of vegetable fibers, the greater the number of cellulose molecules bounded on this way (known as a polysacaride) the greater the resulting fibers will be and, as we saw previously, this is vital for paper strength. The cotton rag that surrounds the fruit has the purest vegetable cellulose fibers in nature and they can hold until 6000 molecules in a polysacaride. This bound is so strong that only herbivore's digesting systems can break. As an interesting fact cotton fibers can size up to 3 cm when pine wood fibers only 3 mm.

Normally first wall on plant's cells contain pure cellulose and when going deeper inside different components appears, as lignine. Next table (sorry, in spanish) shows the % contents of different types of plants:


Source: TAPPI

all this extra elements affect badly to paper's quality and lifetime and different procedures has been created to extract them from the plant, many of them chemical and heavily toxic to the environment. That's the reason why paper factories are eyestorms for nature's defending groups all over the world, since this factories throw their industrial waste normally to rivers or seacoasts, together with the fact that they change natural flora by pine trees, which roots have an strong factor of fertile earth's destruction but a shorter and cheaper time of growing and development as adult trees.

As we said, in 1852, american Benjamin Chew Tilghman patented the method to obtain cellulose from wood, by the chemical creation of cellulose sulfithe. In the world, main trend is to obtain cellulose from this source (woods), except for India (only a 40%) and China, which obtains 80% of it from other plants [1].

Finally to say that cellulose molecular bound is so strong and closed that paper is more rigid in one direction than the other, and as LaFosse says in his book if you hold a square by one edge it inclines down lower than if you hold it by the next edge on its side, because fibers trends to align in a direction. Also cellulose in insoluble on water and, when water is added to paper, fibers grow wider, not longer, and that's the reason because paper doesn't get bigger and wrinkles appears on its surface (an important fact when you try wet folding).

Beating



In Part I we saw that long fibers creates a resistant but rough paper and short fibers a softer and fine one. The chemical process to separate cellulose from plant leaves untouched and separated fibers, so if you want to get fineness it is necessary then a beating and agitating stage. This is achieved by the usage of mills and pulp circuits, most common of them is the Hollander mill (image above).

It is basically a water-pulp circuit plus a rotating mill with square gears that reduces the passage area of the liquid, beating the fibers, cutting and spreading them in the solution.

As long the time the pulp is beated, the greater its opacity and fineness of the resulting paper, also its fragility (normal paper is beated for a couple of hours and tissue paper could be more than 8 hours in the process).

The Pulp Sheet

Once the pulp solution leaves the Beating, it is possible to color it by adding retention agents and pigments. Then is necessary to create a layer of homogeneous pulp which will become the sheet of paper. One way to do it is water mixing and depositing it in a rectangular frame, as cellulose doesn't dissolve in water, it evaporates and leave the pulp layer, of course if you use a frame with a filter screen at the bottom in a way that water drops down the process is accelerated and improved. Industrial paper making spreads the wet pulp in long lines of production which controls its thickness and density.

Pressing

The wet pulp sheet is then pressed according to desired degrees of density required for paper. Here we look again at cellulose fibers since if they are smashed the area covered by them increases and also the pulp density and bounding, making paper with bigger opacity and shapping, as you can see in the next images. Some water is extracted from pulp in this stage also but normally sheets are placed between wet felts.





Driying

Finally paper is driyed by hot air until it keeps its color and natural humidity. Also it is possible to give it some finishing with special surface paints, as for shinny continuous surfaces.

This far we will reach about the processes of paper making, in next posts I will try to talk about the effects on paper of making different types of folds, as long as the effects on stress and recovering of paper and fibers, I hope you are finding this journey interesting so far ;)

Bibliography
1 http://web1.caryacademy.org/chemistry/rushin/StudentProjects/CompoundWebSites/2000/Cellulose/history.htm
2 http://en.wikipedia.org/wiki/Cellulose
3 http://www.forestprod.org/cdromdemo/pf/pf8.html

Her Majesty the Paper (Part I)

Frequently people ask me about what kind of paper I prefer to fold one or another model, if I have a preferred generic one, or if there is a "traditional" or "classic" paper from Japanese origami and what gives it its "special" condition. Then I gave answers like "to complex figures and very detailed is better to use tissue foil ", or "if I want the models to keep their shape in time you can use metalized paper", or "kraft paper is good if you want to wetfold or fold with brush". And everybody get happy thinking how much I know about this handcraft; everybody except me of course who realize that in fact I don't have a clue about what paper is, nor how it works, or why it has this or that quality.

How much knows an origamist about the paper he uses, how it was made, what's the secret that keep it tight and how to select the best for their purposes? And what about the making processes of paper and the ecological problems they mean to nature?

So I decided to do some research.

Maybe a good starting-point should be what paper is not. Word came from latin papirus which signed the plant which were used by the egipcians to make their famous writing rolls .. However, yet the principle is the same, paper we know and use today doesn't have its origins there. Papirus is a plant with long leafs, soft stalk and a triangular wide base and the rolls were made directly from its medulla, a paste which is spread over molds and hydrated with water, pressed and the left dry, to finally be rubbed with ivory or shell to soften its surface. Its origins goes back until 3000 BC and its use s to Greece and the entire Rome Empire until the Vth century. After that, writing was done over parchments, made from fine layers of cow, sheep or ram leather(1).





The true origin for paper lies on China. Around AD 105, emperor Ho Ti ordered his eunuchs chief Tsai Lun the study of new materials for writing, since the wood tablets and silk patches were unpractical for the growing usage of writing. His work concluded with the making of a vegetable pulp made with fibers from bamboo cane, mulberry and other plants, along with the development of a procedure for the making of the paper, which was kept absolutely secret for more than 500 years.

Only after AD 500 the technique of paper making passed to Korea and in AD610, priest Ramjing traveled to Japan for bringing assessment in the production of paper; both countries will upgrade it according to their own resources and technology (in AD700 rice flour was added to the pulp). In 750 it passed to Central Asia, Tibet and India, to finally reach the Arabian world and his vast empire, which ran through all Northern Africa until Spain in Europe.

Here there is an important change in the technique. Arabs, not having many fresh plants, started to use clothe fibers and to recycle materials like old carpets, tapestry or damaged cane products. Pulp then obtained produced a finer paper but with a shorter life; also they incorporated starch, which improved its resistance to the stroke of writing. The first workshop installed in Europe was in the Hispanic Arabian city of Cordoba in AD1036.

Perhaps it is a good moment to start explaining what is paper and how does it work. It is a irregular structure formed by entangled fibers in a paste which is hydrated and left decant in a layer relatively homogeneous. The size of the fibers plays a main part to achieve some properties in the paper; long fibers will give it strength and rigidity, but with a rough finish, and short fibers will produce fine paper, formed, flexible, textured and opaque, but not resistant, better for writing. In mixture of both fibers lies the secret to obtain specific results.

a)raw white paper (1000x), b) secondary cellulose fiber (400x)

secondary cellulose substrate (200x)

With Crusades the manufacturing process arrived to Italy, country where it was incorporated to it a glossy finish with animal grease that gave paper a great surface resistance, which allowed the sharp scribe's pens (made from feather) to write without wrapping it, this made the parchment to disappear very quickly from Europe. The writing technique with feather, dominant in Europe, against the calligraphic brush painted one in the East, determined the final differences between European and Chinese-Japanese paper (2).

With the creation of the printing press on XVth Century, needs of paper grew explosively and clothing resources started to lack, and also hands to do it. In 1798, french Nicholas Louis Robert created the first effective paper machine, which was improved later on 1803 by english brothers Henri and Sealy Fourdrinier. They incorporated on 1840 the crushing of wood to the making process of the pulp. Finally, on 1850, the chemical process to produce pulp was created, which made the production much cheaper. On 1852 Meillier discovered cellulose and Tilghman patented the process to obtain it from wood. Just on 1853, the circle was closed, when paper machines arrived to China and Japan, country that produces the 15% of world's paper needs.

Bibliography (in Spanish):
(1) http://www.papelnet.cl/papel/papel.htm
(2)http://www.papelerapalermo.com/oficios/art-sobre-como-llego-el-papel.asp