Jonathan Livingstone Seagull quotes by Richard Bach

“To begin with, you've got to understand that a seagull is an unlimited idea of freedom"

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"Your whole body, from wingtip to wingtip, is nothing more than your thought itself, in a form you can see. Break the chains of your thought, and you break the chains of your body, too.”

“Don’t believe what your eyes are telling you. All they show is limitation. Look with your understanding. Find out what you already know and you will see the way to fly.”

"We choose our next world through what we learn in this one. Learn nothing, and the next world is the same as this one, all the same limitations and lead weights to overcome."

 “The gull sees farthest who flies highest”

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“Heaven is not a place, and it is not a time. Heaven is being perfect.”

"He was not bone and feather but a perfect idea of freedom and flight, limited by nothing at all"

“To fly as fast as thought, to anywhere that is, you must begin by knowing that you have already arrived.”

 “You don't love hatred and evil, of course. You have to practice and see the real gull, the good in every one of them, and to help them see it in themselves. That's what I mean by love.”
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“Keep working on love.”

“For most gulls it was not flying that matters, but eating. For this gull, though, it was not eating that mattered, but flight.”

“Why is it,” Jonathan puzzled, “that the hardest thing in the world is to convince a bird that he is free, and that he can prove it for himself if he’d just spend a little time practicing? Why should that be so hard?”

“For each of them, the most important thing in living was to reach out and touch perfection in that which they most loved to do...”

Livingstone-seagull-quotes (0-00-21-03)

"The only true law is that which leads to freedom" “Like everything else, Fletcher. Practice.”

Quotes by Richard Bach in Jonathan Livingston Seagull.
Video and Graphics by Diana Matoso.
Audio from


To make a labyrinth what it is, we have to fulfill the following elements:
  • one entrance
  • winding path or path with several turns
  • no crossroads
  • no dead end streets
  • one centre
  • the same path leads to the exit
  • the exit is actually the entrance

Cultural meanings

Prehistoric labyrinths are believed to have served as traps for malevolent spirits or as defined paths for ritual dances. In medieval times, the labyrinth symbolized a hard path to God with a clearly defined center (God) and one entrance (birth). In their cross-cultural study of signs and symbols, Patterns that Connect, Carl Schuster and Edmund Carpenter present various forms of the labyrinth and suggest various possible meanings, including not only a sacred path to the home of a sacred ancestor, but also, perhaps, a representation of the ancestor him/herself: "...many [New World] Indians who make the labyrinth regard it as a sacred symbol, a beneficial ancestor, a deity. In this they may be preserving its original meaning: the ultimate ancestor, here evoked by two continuous lines joining its twelve primary joints."
One can think of labyrinths as symbolic of pilgrimage; people can walk the path, ascending toward salvation or enlightenment. Many people could not afford to travel to holy sites and lands, so labyrinths and prayer substituted for such travel. Later, the religious significance of labyrinths faded, and they served primarily for entertainment, though recently their spiritual aspect has seen a resurgence.
Walking among the turnings, one loses track of direction and of the outside world, and thus quiets the mind.

Spiritual Walks

They are energy grids that have been around over the centuries, paths for personal, psychological and spiritual transformation.

To traverse the labyrinth is a journey into the center of our own being and the return to our divine source. Each single round of a labyrinth is called a circuit.

How to Walk a Labyrinth

The labyrinth embodies a “unicursal” design: It has one continuous path. The path in is the path out. There is no wrong way to walk it. There are no dead ends. You can’t get lost.
Before you enter, allow each part of your body to relax. Breathe slowly and deeply.  As you approach your walk, release your concerns, and be open to receiving. Consider bringing an issue or a question to your walk.  This allows you to focus the intention of your walk and receive guidance. To walk the path is to know and trust that guidance is available to you. Allow yourself to be who you are and to walk in whatever way is best for you. Your experience will be yours, and whatever you experience, know that it is right for you.
Among the meander labyrinths, there are 2 main designs, the seven-circuit type and the medieval design. The seven-circuit ones are also known as Cretan Labyrinths.
Traveling the seven-circuit labyrinth is extremely powerful, as it correlates to the seven chakras:
1. Base chakra (Root). Your relationship to the universe
2. Sacral chakra (Navel). Your relationship to the community in which you exist
3. Solar plexus. Your perception of yourself
4. Heart. The point of oneness in the human body with the all; the balance between the lower and higher chakras
5. Throat. How you speak your truth
6. Third Eye. How you see the truth
7. Crown. Your spiritual relation to the universe
If you walk a classical 7-circuit labyrinth, the following pattern emerges:
If you work with the chakras, it indicates that you go through the heart (4) from the material world (1, 2, 3) to the spiritual world (5, 6, 7). The heart is in the centre. This strengthens the knowing that love is all there is. Love is the only truth, the only reality, the only power. Love is eternal.
If you draw lines in the sequence that you walk the labyrinth; that is 3-2-1-4-7-6-5, it forms what looks like a cup. This cup, some say, relates to the Holy Grail and its secret knowledge. The wisdom lies within.
Your heart is known by the path you walk.

Labyrinth VS Maze

Labyrinth is indeed a complex and confusing structure, but it is not a maze! Unfortunately, only a few people know about this difference. Therefore, in general use, we have the Platonic function of the labyrinths as a synonym for confusion and disorientation.
Look at the picture of a simple maze.
As you can see, it has an entrance and an exit on two opposite sides. As soon as you enter, you face a dilemma: to head on straight or turn left. If you turn left you will find yourself in a dead end street and you will have to go back. In some places there are crossroads where you can choose whether to go one way or the other. In this maze there is only one correct path that leads to the exit. However, it is possible to construct a maze whose paths connect, meaning there can be alternative paths that lead to the exit. If this maze was more complex, one could spend hours in it before finding a way out.
The popular term ‘labyrinth’ is often wrongly used to describe these kinds of structures (in parks and similar). As I have pointed out, a maze significantly differs from the definition of a real labyrinth.
Here is how a labyrinth looks like.
This is also a complex and a confusing structure, at least when you first look at it. Still, if you examine it more closely you will discover it only has one entrance (which is also an exit), so there are no crossroads or dead end paths. This means that after you reach the centre you go back the same way to reach the exit.
As much fun and likable mazes can be, they are not labyrinths. While mazes symbolize the process of deciding and reaching decisions, stimulating the mind to get around (the body, also if they are the right size), labyrinths are true patterns of power that work on a much deeper level than pure logic and getting around. 

Drawing a Labyrinth by Yourself

The 7-circuit labyrinth is drawn as shown. To draw an 11-circuit labyrinth, add an "L" in each corner and follow the same plan as for 7-circuit -- see diagram, lower right.


Global Brain

Global Brain

Here's an image we all know well . . its an image which is in many ways a symbol of our times . . It represents what is perhaps our greatest technological achievement so far - the step out into space. But the picture of our planet also has a far deeper significance . . . for the first time we have seen ourselves and the planet from the outside . . and the view has changed our feelings about ourselves in an unexpected way.

Image Source: Nasa

For the early astronauts, the sight of the planet, shining in the blackness of space, was a profound experience.
One of the astronauts who walked on the moon, described his view of the earth as an experience of instant global consciousness. He said "When you are up there you're no longer an American citizen, or a Russian citizen, suddenly all those boundaries disappear, you're a planetary citizen".
Similar experiences happened to the other astronauts. They saw Earth to be a very beautiful, magical-looking planet, and came back deeply changed from that experience. . . Moreover, they became increasingly aware that all was not well back on earth, and they came back wanting to help in some way or another. . . The pictures they returned with also had a deep effect on many of us back on Earth . . . For many people it's an image which brings forth feelings of wonder, and of awe. . . of feelings of connection . . of being part of something greater.

 In some respects the picture of our planet is a spiritual symbol of our times. Now that may initially sound strange, what is spiritual about it? But, if you think for a moment, most spiritual symbols are, in one way or another, symbols of wholeness, of oneness, of some underlying unity. And a sense of oneness and unity is exactly what is reflected in this picture of the planet. . It symbolises the growing awareness that we are One Humanity, living on One Planet, and with One Common Destiny. This is why the picture of the planet is so powerful, . . and why it has become such an important symbol for all those who are concerned about the fate of humanity, and creating a better world. And there was another awesome realisation which struck some of the early astronauts. Looking back at the earth, it seemed that maybe the whole planet is a single living organism. Until we took the step out into space, we had only seen the planet from close up. Even when you're ten miles up in a plane you're still only seeing a very tiny part of it. In some respects we could compare ourselves to fleas living on an elephant. Now, being scientific fleas, they have studied the skin, measured how deep it is, looked for the sweaty bits, found the hairy bits, and generally mapped out the elephant. Then one day, one of these fleas takes a huge leap off - 50ft. away - and, looking back at the elephant, wonders, "Could this elephant actually be a living organism in its own right?" We have been taking a similar huge leap back - from 100 miles . . . to 1000 miles . . till eventually we see the planet as a single entity. And the question comes - could planet earth be a single living organism? Now this is not such a wild dream as you might suppose. A number of scientists are now beginning to take the idea quite seriously. Researchers in both England and America have been looking at some of the long term changes which have happened to the planet. When,for example, you look at how the temperature of the planet has varied over the whole of its history, you find that it started off very hot, but then cooled rapidly . . until it reached the crucial range in which life could evolve. From then on it seems to have remained steady. Now that's strange because theoretically it should sooner or later have started rising again, becoming far too hot for life to continue. But it didn't. It stayed constant. what seems to have happened is that the living organisms of the time released the right amounts of the right gases to keep the temperature of the atmosphere constant . . at just right the right level for life to continue evolving. That may sound an amazing coincidence - but it is no more amazing than the fact that our own bodies sweat when we are hot, and shiver when we are cold, maintaining ourselves at a constant temperature. The salt content of the sea also shows an interesting degree of self-regulation. Salt is continually washed of the land into the sea, and at such a rate that it would take only eighty million years for the oceans to reach their present concentration. So theoretically the oceans should by now have become one huge dead sea. But they haven't. In fact the salt content has remained remarkably steady. Somehow small organisms in the sea have absorbed just the right amount of salt to keep it constant.
These are just two of many examples which have led scientists to propose that all the living organisms on the planet together function as one huge self-regulating system -- a theory known as the Gaia Hypothesis - GAIA - after this lady, who is the Greek Earth Mother Goddess - Gaia.
We might then, consider our days and nights to be like the heartbeat of Gaia . . . the seasons to be her breaths . . the tropical rain forests resemble her lungs . . and the oceans are like the circulatory system. So, if the whole planet does behave like a huge living system, "what", we might then ask, "are we doing here?" What is the humanity's function in this system? To get some perspective lets go back and look at how we came to be here . . at the whole evolutionary process which has led to us. According to current scientific theory the Universe started with a big bang - pure energy, light - in the beginning there was light - something science and religion actually agree upon. Then this energy quickly condensed into electrons and protons - the first elementary particles. And then these very simple particles began combining to form the first simple atoms. Which themselves began collecting together to form more complex atoms . . . and more complex atoms . . until the whole of chemistry had evolved. Next these different atoms came together to form molecules - like this amino acid, one of the basic building blocks of life. These, over time, began stringing themselves together to form much more complex molecules, like DNA, containing millions of atoms. These molecules then grouped themselves to form the first simple living organisms . . bacteria and algae. Over time these too collected together to form multi-cellular organisms . . simple sponges, and the first plants. This process of progressive complexification and organisation continued, leading eventually to the first fish. Now fish represent a very significant stage in evolution. Fish were the first creatures with a skull and a spinal cord. And inside was the delicate nervous system - which now, for the first time in history, was protected by a case of bone. And from then on the main thrust of evolution has been in the development of the nervous system. As we move on up through evolution, the external changes are very clear. The gills have become lungs, the fins have changed into arms and legs . . But the most important changes of all were happening on the inside. The nervous system has become unfathomably complex, and today, the most complex nervous systems known on the planet are to be found in dolphins and whales. The fact that they have more complex brains than our own, suggests they might be more intelligent than us - though what they do with that increased capacity we do not as yet really know. Leaving them aside for now, let us consider the second most complex nervous system on this planet - the human nervous system. One of the unique attributes of the human mind is our ability to study the world around, and understand it. It is this which underlies our science, our technology, and the whole of human culture. . . . We are, as we have seen, the product of this long evolutionary journey, through the evolution of matter and the stars, through the evolution of life and simple organisms, to the stage where now, with us, evolution has reached the crucial stage of complexity whereby it can look back on the Universe . . Whether we are studying a stone, a leaf or looking out into the heavens we are the Universe's way of beginning to explore itself. . . We are a stars' way of exploring the stars. . . And the same could be said of our inner reflections.
When we turn our attention inwards and begin to explore our own minds, and discover deeper levels within ourselves, we are the Universe beginning to investigate its inner dimensions.
So the next question is, "Where is all this taking us?" What's next? In the long term, over millions of years, we will probably see various physical changes, such as our brains growing even larger; but there are other evolutionary developments happening a lot more quickly. One of the dominant trends of evolution has been a progressive linking of smaller units into larger and larger units. This suggests that the next stage might involve human beings themselves linking together in some way, indeed this process has already begun - it is what we call civilisation. . . The organisation of the streets in this Danish village is very similar to the the structure of some of the first multi- cellular organisms in the sea. This is Sun City, Arizona - the pattern here is similar to a growing foetus. And this tendency of people coming together into large units is also happening at a much higher level. We can see it in the union of the States in the U.S.A. and the USSR, and the coming together of the EEC in Europe. This is the natural drift of evolution, and its end result would seem to be the whole of humanity beginning to function as one single community. As well as this linking at a physical level, we are also beginning to link together at a deeper level . . As a result of the information explosion, telecommunications, and the birth of computer networks, we are beginning to exchange information with each other wherever we may be . . we are beginning to share our thinking . . we are beginning to connect together mentally. . We are beginning to link up, mind to mind. And, as we begin to understand each other, an even deeper level of linking is occurring - a linking of soul to soul - we are beginning to appreciate our essential unity and oneness. So the next stage of evolution could be humanity beginning to link together physically and mentally. Beginning to work and function on many different levels as an integrated system. Now there are some interesting parallels between this integration of humanity and the integration which occurred at earlier stages of evolution. If we go back and consider the emergence of life from matter, we find that the simplest bacteria contain several billion atoms. The figure shown here of ten billion is only a very approximate figure - whether its 2 billion, 5 billion or 20 billion doesn't really matter. But it seems necessary to bring together approximately this number of atoms in order to create the complexity necessary for them to become a living system. Similarly, if we look at the next leap in evolution, the evolution of intelligent consciousness, we find it again takes several billion nerve cells linked together in the human brain to produce the reflective consciousness characteristic of humanity - though again the figure is not meant to be exact. Now, if this turns out to be a general principal of evolution, it would suggest that the next stage - the linking of humanity into an integrated system - would involve the working together of a similar number of minds - and indeed the human population has already reached this level. We might then think of humanity as some sort of huge global brain - a brain in which we are the cells linked together by our growing information networks. And there, indeed, are some interesting parallels between the way society is developing and the way the human brain develops. The main thrust of development of the human brain takes place mainly before birth - between weeks 8 and 13 after conception. Imagine for a moment that you are a nerve cell in this growing human brain. At first there's plenty of space . . Then very quickly there's a massive population explosion of nerve cells. . . If you were a nerve cell you'd probably think "This is getting dangerous. There's not enough oxygen to go round, we're going to be short of blood soon. . . But suddenly, at week 13, the explosion stops. From then on the development of the nervous system focuses on the growth of connectivity and complexity . . the linking together of these billions of nerve cells. Today we are seeing a similar process happening to humanity. We've had this massive population explosion, but its now beginning to slow down, and we seem to be moving into the next phase, the linking of the billions of human cells in this planetary brain. Through postal systems, telephones, computer networks, and satellites, we are increasing the connectivity and linkage of the billions of minds which together constitute humanity. Now all that I've spoken of so far is the good news. . the bad news is that our rapid development is also threatening the welfare of the planet. In some respects we seem to be very much like a planetary cancer. Its interesting that if you take a photograph of a city from the air, and look at the way the city spreads out into the environment, its very reminiscent of the way a cancer grows in the body. . . This, for example, may look like Los Angeles from the air, but its not. Its actually a cancerous tissue in the human body. But take a good look at it - notice how it is growing . . . and then compare it with this. Now the similarity is more than just in appearance - if you go back and look at what causes the cancerous tendency in the individual, and at what causes a similar tendency in society, we find that underneath are very similar principles. Lets look first at why a cell in the body becomes cancerous. In the centre of each cell are the genes. They contain the information that keep you functioning as a single living organism, rather than just a bowl of biological soup. Now if the genes in a cell are disturbed, that cell may become selfish, . . it may no longer support the system as a whole, but instead go off, doing its own thing, at the expense of the body , - - it becomes a cancer. Now when we consider human beings in a community, we are looking at an organisation of minds. And the parallel to the genes is now to be found in the centre of minds, at our inner cores. Its that deepest level of inner wisdom which many mystics and philosophers have often spoken of. That inner awareness of being much more than we normally experience - a part of something much greater. and when we loose touch with this inner wisdom we also become selfish cells, out of touch with the needs of society as a whole, living at the expense of each other.
20 Amazing Facts About The Human Body That Will Blow Your Mind.Image Source: Viralcy

 The philosopher Alan Watts referred to this selfish isolated way of existence as "the skin encapsulated ego". What's inside the skin is me - and what's outside the skin is not me. Biologically speaking this is, of course, true - we are each separate biological individuals. But it is not the whole truth. We are much more than that . . we are creatures with an inner life - with an existence that stretches beyond our biological identity.
We become stuck in this limited way of seeing ourselves because the real self, our deepest sense of I" - what some call the pure self - is actually very hard to grasp. Trying to describe that deeper sense of self is very much like trying to describe a hole in a piece of wood. If you ask people to describe a hole such as this, they may start by saying "Well, its a round hole", you say "Yes", they say "It's a wooden hole, and its red". But you say "Hang on, the hole isn't wood, the hole isn't red. And they say "Ah-ha the hole is black", "No, that's the background". And suddenly they're stuck - what's the hole? How do you describe the hole itself, without describing its surroundings?
In a similar way, it is very difficult to grasp and define our own inner sense of self. Instead we tend to describe our selves in terms of what surrounds us. . . our many possessions . . the roles we play, our social status . . our beliefs - both scientific and religious. . . This limited sense of identity may not in itself seem very dangerous, but it does have some far- reaching consequences. . . It turns out that many of the ways in which we mistreat and abuse the environment stem from our seeing the world as separate from ourselves. We may take fairly good care of what is inside the skin, but we don't care nearly so well for what is outside the skin. As the late Gregory Bateson said; "If this, meaning this me versus the world attitude, if this is your estimate of your relationship to nature, and you have advanced technology, your likelihood of survival will be that of a snowball in hell. You will die either of the toxic by-products of your own hate, or simply, of over-population and over-grazing". And Bateson went on to say "That if I'm right the whole of our thinking of what we are, what other people are, has got be be restructured. The most important task today is to learn to think in a new way". I would actually go a step further and say "The most important task is to be in a new way", to experience, to be conscious in a new way. We need to make the shift from this - the skin encapsulated model of the self . . . to this . . . what some have referred to as "leaky margins". The boundaries are still there, but they are much less solid. In addition we now experience a greater oneness with the world outside. Such a shift of consciousness could play an important part in the next stage of our evolution. If we go back and look at the previous major steps in evolution, we can see that each stage became a platform for the next stage - energy led to matter, matter led to life, and now life has led to consciousness. Thus the spearhead of evolution is now the human mind. We have moved beyond biological evolution into mental evolution. Thus it is not changes in our bodies which will now determine the future, so much as changes in our thinking, in our perception, and in our attitudes. The evolutionary phase which we have now entered is the evolution of human consciousness - inner development is the key . . . In short, evolution has now become internalised. And these changes are happening much faster than we might at first suspect. Recent data shows that inner evolution may soon become very widespread in our society. . . If we look back at the past distribution of human employment in the West, we see that the major focus for centuries was agriculture . . . most people were employed on the land. That continued up until 1900 when a new curve took over - industry. Industrial work became the dominant use of human time. We entered "the industrial age". And the industrial age continued until 1975 when a new curve took over, information processing , and we are now very much in the information age. We've entered into the information age much faster than anybody anticipated . . . whereas it took over a century for the Industrial Revolution to have its full effect, the Information Revolution has had a major impact in just twenty five years. So the next question is; What will follow the information age . ? ? Well, there are already signs of a new curve beginning, and although small at present it is growing even faster than information processing . . This curve represents the growing number of people involved in the exploration and development of the vast resources of the human mind itself. These are the people who are beginning to explore what may perhaps be the last great frontier - our own minds. The number may be small at present, but as you can see, if you extrapolate the trend, then by the year 2000 it will have overtaken the information curve. What we would then see is a very major shift in values, as more and more people discover a deeper sense of unity and purpose . . and, letting go of their petty selfish ways, beginning to function more in tune with humanity and the planet as a whole. Now the idea that there are deeper levels to human consciousness is not new. In all ages, all around the world, we find people talking of that same fundamental wisdom, that same basic understanding of life and consciousness. Here I've symbolised these different people at different times, by different coloured dots. But when any particular teacher died, his message was forced to spread by word of mouth, or be written down on parchments, with the result that as his ideas began to spread around the world, they inevitably became distorted and absorbed by the culture of the time, with the result that very little now remains of the original wisdom. But today, we are in very different circumstances. Firstly, hundreds upon thousands of people are beginning to discover the hidden potentials of the human soul. . . And secondly, books, tapes, television, and computer networks - in short, the whole information revolution - has given us the means to share these discoveries with each other without the information suffering the distortion and misunderstanding that was inevitable in the past. So now, as the wisdom begins to spread out across the planet, there is instead a positive feedback - the wisdom reinforces itself . . . rather than dissolving the wisdom builds upon itself . . . This inner awakening could be the crucial ingredient in the linking of humanity into an integrated society , a society in which we are linked spiritually through an awareness of our inner unity . . an awareness that we are all part of something greater . . and at the same time gaining a greater awareness of our individual potentials and uniqueness. . . a synthesis of greater individuality along with greater community. Such an inner development, if widespread, could be very valuable in helping humanity deal with the problems now facing it. We are clearly in a time of crisis.
Most of us think of crises as bad; but maybe we have something to learn from the chinese on this. Their word for crisis, Wei Chi, means two things. The first symbol means danger, beware. But the other two symbols mean opportunity - an opportunity for something new to emerge . . .
If we only see crisis as danger, as a threat to our accustomed ways, then we may spend all our energy resisting it. We need also to look at the opportunities inherent in the crises. From an evolutionary perspective crises are a sign that something has gone wrong . . the patterns of the past are no longer working . . . Crises are thus a challenge. The challenge to adapt. They are the challenge to let go of the old ways of thinking and move on to a new way of seeing. . . Humanity's current crisis may not, at its root, be an economic crisis or an environmental crisis - it may well be a crisis of consciousness, a crisis in how see ourselves, and the world around. Many of us have probably experienced at one time or another those moments when we feel at one with the world; a sense of inner peace, with no need to prove who we are. The question is, can we allow these precious moments to happen more often in our lives - and in the lifes of other people. In short, can we choose to explore and develop our greatest resource of all - our own minds, our inner selves. I'd like to finish with a poem from Christopher Fry's play "A sleep of Prisoners". The human heart can go the length of God, Dark and cold we may be, but this is no winter now. The frozen misery of centuries cracks, breaks, begins to move, The thunder is the thunder of the floes, The thaw, the flood, the upstart Spring, Thank God our time is now, When wrong comes up to meet us everywhere, Never to leave us till we take, the greatest stride of soul man ever took. Affairs are now soul size, The enterprise is exploration into God, But where are you making for, It takes so many thousand years to wake, but will you wake, for pity's sake......

Transcript from the The Global Brain by Peter Russel

Rarefractions [definition & visual research]

A decrease in the density of something is rarefaction. As you climb a mountain, you experience rarefaction of the air; the air becomes less dense the higher up you go.
Most of the time, rarefaction refers to air or other gases becoming less dense. When rarefaction occurs, the particles in a gas become more spread out. You may come across this word in the context of sound waves. A sound wave moving through air is made up of alternating areas of higher and lower density. The areas of lower density are called rarefactions. [*]

in the physics of sound, segment of one cycle of a longitudinal wave during its travel or motion, the other segment being compression. If the prong of a tuning fork vibrates in the air, for example, the layer of air adjacent to the prong undergoes compression when the prong moves so as to squeeze the air molecules together. When the prong springs back in the opposite direction, however, it leaves an area of reduced air pressure. This is rarefaction. A succession of rarefactions and compressions makes up the longitudinal wave motion that emanates from an acoustic source. [1]

Gears [research]

gear or cogwheel is a rotating machine part having cut teeth, or cogs, which mesh with another toothed part to transmit torque, in most cases with teeth on the one gear being of identical shape, and often also with that shape on the other gear. Two or more gears working in tandem are called a transmission and can produce a mechanical advantage through a gear ratio and thus may be considered a simple machine. Geared devices can change the speed, torque, and direction of a power source. The most common situation is for a gear to mesh with another gear; however, a gear can also mesh with a non-rotating toothed part, called a rack, thereby producing translation instead of rotation.

The gears in a transmission are analogous to the wheels in a crossed belt pulley system. An advantage of gears is that the teeth of a gear prevent slippage.
When two gears mesh, and one gear is bigger than the other (even though the size of the teeth must match), a mechanical advantage is produced, with the rotational speeds and the torques of the two gears differing in an inverse relationship.
In transmissions with multiple gear ratios—such as bicycles, motorcycles, and cars—the term gear, as in first gear, refers to a gear ratio rather than an actual physical gear. The term describes similar devices, even when the gear ratio is continuous rather than discrete, or when the device does not actually contain gears, as in a continuously variable transmission.[1]


History of the differential gear[edit]

The earliest known reference to gears was circa A.D. 50 by Hero of Alexandria,[2] but they can be traced back to the Greek mechanics of the Alexandrian school in the 3rd century BCE and were greatly developed by the Greek polymath Archimedes (287–212 BCE).[3] The Antikythera mechanism is an example of a very early and intricate geared device, designed to calculate astronomical positions. Its time of construction is now estimated between 150 and 100 BC.[4]

Single stage gear reducer.
Ma Jun (c. 200–265 AD) re-invented the differential gear as part of a south-pointing chariot.

History of other gears[edit]

Comparison with drive mechanisms[edit]

The definite velocity ratio that teeth give gears provides an advantage over other drives (such as traction drives and V-belts) in precision machines such as watches that depend upon an exact velocity ratio. In cases where driver and follower are proximal, gears also have an advantage over other drives in the reduced number of parts required; the downside is that gears are more expensive to manufacture and their lubrication requirements may impose a higher operating cost.


External vs internal gears[edit]

Internal gear
An external gear is one with the teeth formed on the outer surface of a cylinder or cone. Conversely, an internal gear is one with the teeth formed on the inner surface of a cylinder or cone. For bevel gears, an internal gear is one with the pitch angle exceeding 90 degrees. Internal gears do not cause output shaft direction reversal.[5]


Spur gear
Spur gears or straight-cut gears are the simplest type of gear. They consist of a cylinder or disk with the teeth projecting radially, and although they are not straight-sided in form (they are usually of special form to achieve constant drive ratio, mainly involute), the edge of each tooth is straight and aligned parallel to the axis of rotation. These gears can be meshed together correctly only if they are fitted to parallel shafts.


Helical gears
Top: parallel configuration
Bottom: crossed configuration
Helical or "dry fixed" gears offer a refinement over spur gears. The leading edges of the teeth are not parallel to the axis of rotation, but are set at an angle. Since the gear is curved, this angling causes the tooth shape to be a segment of a helix. Helical gears can be meshed in parallel or crossed orientations. The former refers to when the shafts are parallel to each other; this is the most common orientation. In the latter, the shafts are non-parallel, and in this configuration the gears are sometimes known as "skew gears".
The angled teeth engage more gradually than do spur gear teeth, causing them to run more smoothly and quietly.[6] With parallel helical gears, each pair of teeth first make contact at a single point at one side of the gear wheel; a moving curve of contact then grows gradually across the tooth face to a maximum then recedes until the teeth break contact at a single point on the opposite side. In skew gears, teeth suddenly meet at a line contact across their entire width causing stress and noise. Skew gears make a characteristic whine at high speeds. Whereas spur gears are used for low speed applications and those situations where noise control is not a problem, the use of helical gears is indicated when the application involves high speeds, large power transmission, or where noise abatement is important.[7] The speed is considered to be high when the pitch line velocity exceeds 25 m/s.[8]
A disadvantage of helical gears is a resultant thrust along the axis of the gear, which needs to be accommodated by appropriate thrust bearings, and a greater degree of sliding friction between the meshing teeth, often addressed with additives in the lubricant.

Skew gears[edit]

For a 'crossed' or 'skew' configuration, the gears must have the same pressure angle and normal pitch; however, the helix angle and handedness can be different. The relationship between the two shafts is actually defined by the helix angle(s) of the two shafts and the handedness, as defined:[9]
E = \beta_1 + \beta_2 for gears of the same handedness
E = \beta_1 - \beta_2 for gears of opposite handedness
Where \beta is the helix angle for the gear. The crossed configuration is less mechanically sound because there is only a point contact between the gears, whereas in the parallel configuration there is a line contact.[9]
Quite commonly, helical gears are used with the helix angle of one having the negative of the helix angle of the other; such a pair might also be referred to as having a right-handed helix and a left-handed helix of equal angles. The two equal but opposite angles add to zero: the angle between shafts is zero—that is, the shafts are parallel. Where the sum or the difference (as described in the equations above) is not zero the shafts are crossed. For shafts crossed at right angles, the helix angles are of the same hand because they must add to 90 degrees.

Double helical[edit]

Double helical gears
Main article: Double helical gear
Double helical gears, or herringbone gears, overcome the problem of axial thrust presented by "single" helical gears, by having two sets of teeth that are set in a V shape. A double helical gear can be thought of as two mirrored helical gears joined together. This arrangement cancels out the net axial thrust, since each half of the gear thrusts in the opposite direction resulting in a net axial force of zero. This arrangement can remove the need for thrust bearings. However, double helical gears are more difficult to manufacture due to their more complicated shape.
For both possible rotational directions, there exist two possible arrangements for the oppositely-oriented helical gears or gear faces. One arrangement is stable, and the other is unstable. In a stable orientation, the helical gear faces are oriented so that each axial force is directed toward the center of the gear. In an unstable orientation, both axial forces are directed away from the center of the gear. In both arrangements, the total (or net) axial force on each gear is zero when the gears are aligned correctly. If the gears become misaligned in the axial direction, the unstable arrangement generates a net force that may lead to disassemble of the gear train, while the stable arrangement generates a net corrective force. If the direction of rotation is reversed, the direction of the axial thrusts is also reversed, so a stable configuration becomes unstable, and vice versa.
Stable double helical gears can be directly interchanged with spur gears without any need for different bearings.


Main article: Bevel gear

Bevel Gear
A bevel gear is shaped like a right circular cone with most of its tip cut off. When two bevel gears mesh, their imaginary vertices must occupy the same point. Their shaft axes also intersect at this point, forming an arbitrary non-straight angle between the shafts. The angle between the shafts can be anything except zero or 180 degrees. Bevel gears with equal numbers of teeth and shaft axes at 90 degrees are called mitre gears.

Spiral bevels[edit]

Spiral bevel gears
Main article: Spiral bevel gear
Spiral bevel gears can be manufactured as Gleason types (circular arc with non-constant tooth depth), Oerlikon and Curvex types (circular arc with constant tooth depth), Klingelnberg Cyclo-Palloid (Epicycloide with constant tooth depth) or Klingelnberg Palloid. Spiral bevel gears have the same advantages and disadvantages relative to their straight-cut cousins as helical gears do to spur gears. Straight bevel gears are generally used only at speeds below 5 m/s (1000 ft/min), or, for small gears, 1000 r.p.m.[10]
Note: The cylindrical gear tooth profile corresponds to an involute, but the bevel gear tooth profile to an octoid. All traditional bevel gear generators (like Gleason, Klingelnberg, Heidenreich & Harbeck, WMW Modul) manufactures bevel gears with an octoidal tooth profile. IMPORTANT: For 5-axis milled bevel gear sets it is important to choose the same calculation / layout like the conventional manufacturing method. Simplified calculated bevel gears on the basis of an equivalent cylindrical gear in normal section with an involute tooth form show a deviant tooth form with reduced tooth strength by 10-28% without offset and 45% with offset [Diss. Hünecke, TU Dresden]. Furthermore those "involute bevel gear sets" causes more noise.


Hypoid gear
Hypoid gears resemble spiral bevel gears except the shaft axes do not intersect. The pitch surfaces appear conical but, to compensate for the offset shaft, are in fact hyperboloids of revolution.[11][12] Hypoid gears are almost always designed to operate with shafts at 90 degrees. Depending on which side the shaft is offset to, relative to the angling of the teeth, contact between hypoid gear teeth may be even smoother and more gradual than with spiral bevel gear teeth, but also have a sliding action along the meshing teeth as it rotates and therefore usually require some of the most viscous types of gear oil to avoid it being extruded from the mating tooth faces, the oil is normally designated HP (for hypoid) followed by a number denoting the viscosity. Also, the pinion can be designed with fewer teeth than a spiral bevel pinion, with the result that gear ratios of 60:1 and higher are feasible using a single set of hypoid gears.[13] This style of gear is most common in driving mechanical differentials, which are normally straight cut bevel gears, in motor vehicle axles.


Crown gear
Main article: Crown gear
Crown gears or contrate gears are a particular form of bevel gear whose teeth project at right angles to the plane of the wheel; in their orientation the teeth resemble the points on a crown. A crown gear can only mesh accurately with another bevel gear, although crown gears are sometimes seen meshing with spur gears. A crown gear is also sometimes meshed with an escapement such as found in mechanical clocks.


Worm gear

4-start worm and wheel
Main article: Worm drive
Main article: Slewing drive
Worm gears resemble screws. A worm gear is usually meshed with a spur gear or a helical gear, which is called the gearwheel, or worm wheel.
Worm-and-gear sets are a simple and compact way to achieve a high torque, low speed gear ratio. For example, helical gears are normally limited to gear ratios of less than 10:1 while worm-and-gear sets vary from 10:1 to 500:1.[14] A disadvantage is the potential for considerable sliding action, leading to low efficiency.[15]
A worm gear is a species of helical gear, but its helix angle is usually somewhat large (close to 90 degrees) and its body is usually fairly long in the axial direction. These attributes give it screw like qualities. The distinction between a worm and a helical gear is that least one tooth persists for a full rotation around the helix. If this occurs, it is a 'worm'; if not, it is a 'helical gear'. A worm may have as few as one tooth. If that tooth persists for several turns around the helix, the worm appears, superficially, to have more than one tooth, but what one in fact sees is the same tooth reappearing at intervals along the length of the worm. The usual screw nomenclature applies: a one-toothed worm is called single thread or single start; a worm with more than one tooth is called multiple thread or multiple start. The helix angle of a worm is not usually specified. Instead, the lead angle, which is equal to 90 degrees minus the helix angle, is given.
In a worm-and-gear set, the worm can always drive the gear. However, if the gear attempts to drive the worm, it may or may not succeed. Particularly if the lead angle is small, the gear's teeth may simply lock against the worm's teeth, because the force component circumferential to the worm is not sufficient to overcome friction. (The Sunbeam S7 motorcycle had shaft drive, but instead of using a bevel gear in the rear wheel, the company unwisely specified worm gearing, which proved unreliable and prone to wear).
Worm-and-gear sets that do lock are called self locking, which can be used to advantage, as for instance when it is desired to set the position of a mechanism by turning the worm and then have the mechanism hold that position. An example is the machine head found on some types of stringed instruments.
If the gear in a worm-and-gear set is an ordinary helical gear only a single point of contact is achieved.[16][13] If medium to high power transmission is desired, the tooth shape of the gear is modified to achieve more intimate contact by making both gears partially envelop each other. This is done by making both concave and joining them at a saddle point; this is called a cone-drive.[17] or "Double enveloping"
Worm gears can be right or left-handed, following the long-established practice for screw threads.[5]


Non-circular gears
Main article: Non-circular gear
Non-circular gears are designed for special purposes. While a regular gear is optimized to transmit torque to another engaged member with minimum noise and wear and maximum efficiency, a non-circular gear's main objective might be ratio variations, axle displacementoscillations and more. Common applications include textile machines, potentiometers and continuously variable transmissions.

Rack and pinion[edit]

Rack and pinion gearing
Main article: Rack and pinion
A rack is a toothed bar or rod that can be thought of as a sector gear with an infinitely large radius of curvature. Torque can be converted to linear force by meshing a rack with a pinion: the pinion turns; the rack moves in a straight line. Such a mechanism is used in automobiles to convert the rotation of the steering wheel into the left-to-right motion of the tie rod(s). Racks also feature in the theory of gear geometry, where, for instance, the tooth shape of an interchangeable set of gears may be specified for the rack (infinite radius), and the tooth shapes for gears of particular actual radii are then derived from that. The rack and pinion gear type is employed in a rack railway.


Epicyclic gearing
Main article: Epicyclic gearing
In epicyclic gearing one or more of the gear axes moves. Examples are sun and planet gearing (see below) and mechanical differentials.

Sun and planet[edit]

Sun (yellow) and planet (red) gearing
Main article: Sun and planet gear
Sun and planet gearing was a method of converting reciprocating motion into rotary motion in steam enginesJames Watt used it on his early steam engines to get around the patent on the crank, but it also provided the advantage of increasing the flywheel speed so Watt could use a lighter flywheel.
In the illustration, the sun is yellow, the planet red, the reciprocating arm is blue, the flywheel is green and the driveshaft is grey.

Harmonic gear[edit]

Harmonic gearing
Main article: Harmonic Drive
harmonic gear is a specialized gearing mechanism often used in industrial motion controlrobotics and aerospace for its advantages over traditional gearing systems, including lack of backlash, compactness and high gear ratios.

Cage gear[edit]

Cage gear in Pantigo Windmill, Long Island (with the driving gearwheel disengaged)
cage gear, also called a lantern gear or lantern pinion has cylindrical rods for teeth, parallel to the axle and arranged in a circle around it, much as the bars on a round bird cage or lantern. The assembly is held together by disks at either end into which the tooth rods and axle are set. Cage gears are more efficient than solid pinions,[citation needed] and dirt can fall through the rods rather than becoming trapped and increasing wear. They can be constructed with very simple tools as the teeth are not formed by cutting or milling, but rather by drilling holes and inserting rods.
Sometimes used in clocks, the cage gear should always be driven by a gearwheel, not used as the driver. The cage gear was not initially favoured by conservative clock makers. It became popular in turret clocks where dirty working conditions were most commonplace. Domestic American clock movements often used them.

Magnetic gear[edit]

All cogs of each gear component of magnetic gears act as a constant magnet with periodic alternation of opposite magnetic poles on mating surfaces. Gear components are mounted with a backlash capability similar to other mechanical gearings. Although they cannot exert as much force as a traditional gear, such gears work without touching and so are immune to wear, have very low noise and can slip without damage making them very reliable.[18] They can be used in configurations that are not possible for gears that must be physically touching and can operate with a non-metallic barrier completely separating the driving force from the load. The magnetic coupling can transmit force into ahermetically sealed enclosure without using a radial shaft seal, which may leak.


Main article: Gear nomenclature

General nomenclature[edit]

Gear words.png
Rotational frequency, n 
Measured in rotation over time, such as RPM.
Angular frequency, ω 
Measured in radians/second1 \mathrm{RPM} = \pi/30 rad/second
Number of teeth, N 
How many teeth a gear has, an integer. In the case of worms, it is the number of thread starts that the worm has.
Gear, wheel 
The larger of two interacting gears or a gear on its own.
The smaller of two interacting gears.
Path of contact 
Path followed by the point of contact between two meshing gear teeth.
Line of action, pressure line 
Line along which the force between two meshing gear teeth is directed. It has the same direction as the force vector. In general, the line of action changes from moment to moment during the period of engagement of a pair of teeth. For involute gears, however, the tooth-to-tooth force is always directed along the same line—that is, the line of action is constant. This implies that for involute gears the path of contact is also a straight line, coincident with the line of action—as is indeed the case.
Axis of revolution of the gear; center line of the shaft.
Pitch point 
Point where the line of action crosses a line joining the two gear axes.
Pitch circle, pitch line 
Circle centered on and perpendicular to the axis, and passing through the pitch point. A predefined diametral position on the gear where the circular tooth thickness, pressure angle and helix angles are defined.
Pitch diameter, d 
A predefined diametral position on the gear where the circular tooth thickness, pressure angle and helix angles are defined. The standard pitch diameter is a basic dimension and cannot be measured, but is a location where other measurements are made. Its value is based on the number of teeth, the normal module (or normal diametral pitch), and the helix angle. It is calculated as:
d= \frac{N m_n}{\cos \psi} in metric units or d= \frac{N}{P_d \cos \psi} in imperial units.[19]
Module or modulus, m 
Since it is impractical to calculate circular pitch with irrational numbers, mechanical engineers usually use a scaling factor that replaces it with a regular value instead. This is known as the module or modulus of the wheel and is simply defined as
where m is the module and p the circular pitch. The units of module are customarily millimeters; an English Module is sometimes used with the units of inches. When the diametral pitch, DP, is in English units,
m=25.4/DP in conventional metric units.
The distance between the two axis becomes
a= m(z_1 + z_2)/2
where a is the axis distance, z1 and z2 are the number of cogs (teeth) for each of the two wheels (gears). These numbers (or at least one of them) is often chosen amongprimes to create an even contact between every cog of both wheels, and thereby avoid unnecessary wear and damage. An even uniform gear wear is achieved by ensuring the tooth counts of the two gears meshing together are relatively prime to each other; this occurs when the greatest common divisor (GCD) of each gear tooth count equals 1, e.g. GCD(16,25)=1; If a 1:1 gear ratio is desired a relatively prime gear may be inserted in between the two gears; this maintains the 1:1 ratio but reverses the gear direction; a second relatively prime gear could also be inserted to restore the original rotational direction while maintaining uniform wear with all 4 gears in this case. Mechanic engineers at least in continental Europe use the module instead of circular pitch. The module, just like the circular pitch, can be used for all types of cogs, not just evolvent based straight cogs.[20]
Operating pitch diameters 
Diameters determined from the number of teeth and the center distance at which gears operate.[5] Example for pinion:
 d_w = \frac{2a}{u+1} = \frac{2a}{\frac{z_2}{z_1}+1}.
Pitch surface 
In cylindrical gears, cylinder formed by projecting a pitch circle in the axial direction. More generally, the surface formed by the sum of all the pitch circles as one moves along the axis. For bevel gears it is a cone.
Angle of action 
Angle with vertex at the gear center, one leg on the point where mating teeth first make contact, the other leg on the point where they disengage.
Arc of action 
Segment of a pitch circle subtended by the angle of action.
Pressure angle\theta 
The complement of the angle between the direction that the teeth exert force on each other, and the line joining the centers of the two gears. For involute gears, the teeth always exert force along the line of action, which, for involute gears, is a straight line; and thus, for involute gears, the pressure angle is constant.
Outside diameter, D_o 
Diameter of the gear, measured from the tops of the teeth.
Root diameter 
Diameter of the gear, measured at the base of the tooth.
Addendum, a 
Radial distance from the pitch surface to the outermost point of the tooth. a=(D_o-D)/2
Dedendum, b 
Radial distance from the depth of the tooth trough to the pitch surface. b=(D-\text{root diameter})/2
Whole depth, h_t 
The distance from the top of the tooth to the root; it is equal to addendum plus dedendum or to working depth plus clearance.
Distance between the root circle of a gear and the dedendum circle of its mate.
Working depth 
Depth of engagement of two gears, that is, the sum of their operating addendums.
Circular pitch, p 
Distance from one face of a tooth to the corresponding face of an adjacent tooth on the same gear, measured along the pitch circle.
Diametral pitch, DP 
Ratio of the number of teeth to the pitch diameter. Could be measured in teeth per inch or teeth per centimeter, but conventionally has units of per inch of diameter. Where the module, m, is in metric units
DP=25.4/m in English units
Base circle 
In involute gears, where the tooth profile is the involute of the base circle. The radius of the base circle is somewhat smaller than that of the pitch circle
Base pitch, normal pitch, p_b 
In involute gears, distance from one face of a tooth to the corresponding face of an adjacent tooth on the same gear, measured along the base circle
Contact between teeth other than at the intended parts of their surfaces
Interchangeable set 
A set of gears, any of which mates properly with any other

Helical gear nomenclature[edit]

Helix angle, \psi 
Angle between a tangent to the helix and the gear axis. It is zero in the limiting case of a spur gear, albeit it can considered as the hypotenuse angle as well.
Normal circular pitch, p_n 
Circular pitch in the plane normal to the teeth.
Transverse circular pitch, p 
Circular pitch in the plane of rotation of the gear. Sometimes just called "circular pitch". p_n=p\cos(\psi)
Several other helix parameters can be viewed either in the normal or transverse planes. The subscript n usually indicates the normal.

Worm gear nomenclature[edit]

Distance from any point on a thread to the corresponding point on the next turn of the same thread, measured parallel to the axis.
Linear pitch, p 
Distance from any point on a thread to the corresponding point on the adjacent thread, measured parallel to the axis. For a single-thread worm, lead and linear pitch are the same.
Lead angle, \lambda 
Angle between a tangent to the helix and a plane perpendicular to the axis. Note that the complement of the helix angle is usually given for helical gears.
Pitch diameter, d_w 
Same as described earlier in this list. Note that for a worm it is still measured in a plane perpendicular to the gear axis, not a tilted plane.
Subscript w denotes the worm, subscript g denotes the gear.

Tooth contact nomenclature[edit]

Point of contact 
Any point at which two tooth profiles touch each other.
Line of contact
A line or curve along which two tooth surfaces are tangent to each other.
Path of action 
The locus of successive contact points between a pair of gear teeth, during the phase of engagement. For conjugate gear teeth, the path of action passes through the pitch point. It is the trace of the surface of action in the plane of rotation.
Line of action 
The path of action for involute gears. It is the straight line passing through the pitch point and tangent to both base circles.
Surface of action 
The imaginary surface in which contact occurs between two engaging tooth surfaces. It is the summation of the paths of action in all sections of the engaging teeth.
Plane of action
The surface of action for involute, parallel axis gears with either spur or helical teeth. It is tangent to the base cylinders.
Zone of action (contact zone) 
For involute, parallel-axis gears with either spur or helical teeth, is the rectangular area in the plane of action bounded by the length of action and the effective face width.
Path of contact
The curve on either tooth surface along which theoretical single point contact occurs during the engagement of gears with crowned tooth surfaces or gears that normally engage with only single point contact.
Length of action
The distance on the line of action through which the point of contact moves during the action of the tooth profile.
Arc of action, Qt 
The arc of the pitch circle through which a tooth profile moves from the beginning to the end of contact with a mating profile.
Arc of approach, Qa 
The arc of the pitch circle through which a tooth profile moves from its beginning of contact until the point of contact arrives at the pitch point.
Arc of recess, Qr 
The arc of the pitch circle through which a tooth profile moves from contact at the pitch point until contact ends.
Contact ratio, mc, ε 
The number of angular pitches through which a tooth surface rotates from the beginning to the end of contact. In a simple way, it can be defined as a measure of the average number of teeth in contact during the period in which a tooth comes and goes out of contact with the mating gear.
Transverse contact ratio, mp, εα 
The contact ratio in a transverse plane. It is the ratio of the angle of action to the angular pitch. For involute gears it is most directly obtained as the ratio of the length of action to the base pitch.
Face contact ratio, mF, εβ 
The contact ratio in an axial plane, or the ratio of the face width to the axial pitch. For bevel and hypoid gears it is the ratio of face advance to circular pitch.
Total contact ratio, mt, εγ 
The sum of the transverse contact ratio and the face contact ratio.
 \epsilon_\gamma = \epsilon_\alpha + \epsilon_\beta
 m_{\rm t} = m_{\rm p} + m_{\rm F}
Modified contact ratio, mo 
For bevel gears, the square root of the sum of the squares of the transverse and face contact ratios.
 m_{\rm o} = \sqrt{m_{\rm p}^2 + m_{\rm F}^2}
Limit diameter 
Diameter on a gear at which the line of action intersects the maximum (or minimum for internal pinion) addendum circle of the mating gear. This is also referred to as the start of active profile, the start of contact, the end of contact, or the end of active profile.
Start of active profile (SAP) 
Intersection of the limit diameter and the involute profile.
Face advance 
Distance on a pitch circle through which a helical or spiral tooth moves from the position at which contact begins at one end of the tooth trace on the pitch surface to the position where contact ceases at the other end.

Tooth thickness nomenclature[edit]

Circular thickness 
Length of arc between the two sides of a gear tooth, on the specified datum circle.
Transverse circular thickness 
Circular thickness in the transverse plane.
Normal circular thickness 
Circular thickness in the normal plane. In a helical gear it may be considered as the length of arc along a normal helix.
Axial thickness
In helical gears and worms, tooth thickness in an axial cross section at the standard pitch diameter.
Base circular thickness
In involute teeth, length of arc on the base circle between the two involute curves forming the profile of a tooth.
Normal chordal thickness
Length of the chord that subtends a circular thickness arc in the plane normal to the pitch helix. Any convenient measuring diameter may be selected, not necessarily the standard pitch diameter.
Chordal addendum (chordal height) 
Height from the top of the tooth to the chord subtending the circular thickness arc. Any convenient measuring diameter may be selected, not necessarily the standard pitch diameter.
Profile shift 
Displacement of the basic rack datum line from the reference cylinder, made non-dimensional by dividing by the normal module. It is used to specify the tooth thickness, often for zero backlash.
Rack shift 
Displacement of the tool datum line from the reference cylinder, made non-dimensional by dividing by the normal module. It is used to specify the tooth thickness.
Measurement over pins 
Measurement of the distance taken over a pin positioned in a tooth space and a reference surface. The reference surface may be the reference axis of the gear, a datum surface or either one or two pins positioned in the tooth space or spaces opposite the first. This measurement is used to determine tooth thickness.
Span measurement 
Measurement of the distance across several teeth in a normal plane. As long as the measuring device has parallel measuring surfaces that contact on an unmodified portion of the involute, the measurement wis along a line tangent to the base cylinder. It is used to determine tooth thickness.
Modified addendum teeth 
Teeth of engaging gears, one or both of which have non-standard addendum.
Full-depth teeth 
Teeth in which the working depth equals 2.000 divided by the normal diametral pitch.
Stub teeth 
Teeth in which the working depth is less than 2.000 divided by the normal diametral pitch.
Equal addendum teeth 
Teeth in which two engaging gears have equal addendums.
Long and short-addendum teeth 
Teeth in which the addendums of two engaging gears are unequal.

Pitch nomenclature[edit]

For other uses, see Pitch.
Pitch is the distance between a point on one tooth and the corresponding point on an adjacent tooth.[5] It is a dimension measured along a line or curve in the transverse, normal, or axial directions. The use of the single word pitch without qualification may be ambiguous, and for this reason it is preferable to use specific designations such as transverse circular pitch, normal base pitch, axial pitch.
Circular pitch, p 
Arc distance along a specified pitch circle or pitch line between corresponding profiles of adjacent teeth.
Transverse circular pitch, pt 
Circular pitch in the transverse plane.
Normal circular pitch, pnpe 
Circular pitch in the normal plane, and also the length of the arc along the normal pitch helix between helical teeth or threads.
Axial pitch, px 
Linear pitch in an axial plane and in a pitch surface. In helical gears and worms, axial pitch has the same value at all diameters. In gearing of other types, axial pitch may be confined to the pitch surface and may be a circular measurement. The term axial pitch is preferred to the term linear pitch. The axial pitch of a helical worm and the circular pitch of its worm gear are the same.
Normal base pitch, pNpbn 
An involute helical gear is the base pitch in the normal plane. It is the normal distance between parallel helical involute surfaces on the plane of action in the normal plane, or is the length of arc on the normal base helix. It is a constant distance in any helical involute gear.
Transverse base pitch, pbpbt 
In an involute gear, the pitch on the base circle or along the line of action. Corresponding sides of involute gear teeth are parallel curves, and the base pitch is the constant and fundamental distance between them along a common normal in a transverse plane.
Diametral pitch (transverse), Pd 
Ratio of the number of teeth to the standard pitch diameter in inches.
 P_{\rm d} = \frac{N}{d} = \frac{25.4}{m} = \frac{\pi}{p}
Normal diametral pitch, Pnd 
Value of diametral pitch in a normal plane of a helical gear or worm.
 P_{\rm nd} = \frac{P_{\rm d}}{\cos\psi}
Angular pitch, θN, τ 
Angle subtended by the circular pitch, usually expressed in radians.
 \tau = \frac{360}{z}  degrees or  \frac{2\pi}{z}  radians


Main article: Backlash (engineering)
Backlash is the error in motion that occurs when gears change direction. It exists because there is always some gap between the trailing face of the driving tooth and the leading face of the tooth behind it on the driven gear, and that gap must be closed before force can be transferred in the new direction. The term "backlash" can also be used to refer to the size of the gap, not just the phenomenon it causes; thus, one could speak of a pair of gears as having, for example, "0.1 mm of backlash." A pair of gears could be designed to have zero backlash, but this would presuppose perfection in manufacturing, uniform thermal expansion characteristics throughout the system, and no lubricant. Therefore, gear pairs are designed to have some backlash. It is usually provided by reducing the tooth thickness of each gear by half the desired gap distance. In the case of a large gear and a small pinion, however, the backlash is usually taken entirely off the gear and the pinion is given full sized teeth. Backlash can also be provided by moving the gears further apart. The backlash of a gear train equals the sum of the backlash of each pair of gears, so in long trains backlash can become a problem.
For situations in which precision is important, such as instrumentation and control, backlash can be minimised through one of several techniques. For instance, the gear can be split along a plane perpendicular to the axis, one half fixed to the shaft in the usual manner, the other half placed alongside it, free to rotate about the shaft, but with springs between the two halves providing relative torque between them, so that one achieves, in effect, a single gear with expanding teeth. Another method involves tapering the teeth in the axial direction and providing for the gear to be slid in the axial direction to take up slack.

Shifting of gears[edit]

In some machines (e.g., automobiles) it is necessary to alter the gear ratio to suit the task, a process known as gear shifting or changing gear. There are several ways of shifting gears, for example:
There are several outcomes of gear shifting in motor vehicles. In the case of vehicle noise emissions, there are higher sound levels emitted when the vehicle is engaged in lower gears. The design life of the lower ratio gears is shorter, so cheaper gears may be used, which tend to generate more noise due to smaller overlap ratio and a lower mesh stiffness etc. than the helical gears used for the high ratios. This fact has been used to analyze vehicle-generated sound since the late 1960s, and has been incorporated into the simulation of urban roadway noise and corresponding design of urban noise barriers along roadways.[21]

Tooth profile[edit]

A profile is one side of a tooth in a cross section between the outside circle and the root circle. Usually a profile is the curve of intersection of a tooth surface and a plane or surface normal to the pitch surface, such as the transverse, normal, or axial plane.
The fillet curve (root fillet) is the concave portion of the tooth profile where it joins the bottom of the tooth space.2
As mentioned near the beginning of the article, the attainment of a nonfluctuating velocity ratio is dependent on the profile of the teeth. Friction and wear between two gears is also dependent on the tooth profile. There are a great many tooth profiles that provides a constant velocity ratio. In many cases, given an arbitrary tooth shape, it is possible to develop a tooth profile for the mating gear that provides a constant velocity ratio. However, two constant velocity tooth profiles have been by far the most commonly used in modern times. They are the cycloid and the involute. The cycloid was more common until the late 1800s; since then the involute has largely superseded it, particularly in drive train applications. The cycloid is in some ways the more interesting and flexible shape; however the involute has two advantages: it is easier to manufacture, and it permits the center to center spacing of the gears to vary over some range without ruining the constancy of the velocity ratio. Cycloidal gears only work properly if the center spacing is exactly right. Cycloidal gears are still used in mechanical clocks.
An undercut is a condition in generated gear teeth when any part of the fillet curve lies inside of a line drawn tangent to the working profile at its point of juncture with the fillet. Undercut may be deliberately introduced to facilitate finishing operations. With undercut the fillet curve intersects the working profile. Without undercut the fillet curve and the working profile have a common tangent.

Gear materials[edit]

Wooden gears of a historicwindmill
Numerous nonferrous alloys, cast irons, powder-metallurgy and plastics are used in the manufacture of gears. However, steels are most commonly used because of their high strength-to-weight ratio and low cost. Plastic is commonly used where cost or weight is a concern. A properly designed plastic gear can replace steel in many cases because it has many desirable properties, including dirt tolerance, low speed meshing, the ability to "skip" quite well [22] and the ability to be made with materials not needing additional lubrication. Manufacturers have employed plastic gears to reduce costs in consumer items including copy machines, optical storage devices, cheap dynamos, consumer audio equipment, servo motors, and printers.

Standard pitches and the module system[edit]

Although gears can be made with any pitch, for convenience and interchangeability standard pitches are frequently used. Pitch is a property associated with linear dimensions and so differs whether the standard values are in the Imperial (inch) or Metric systems. Using inchmeasurements, standard diametral pitch values with units of "per inch" are chosen; the diametral pitch is the number of teeth on a gear of one inch pitch diameter. Common standard values for spur gears are 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 48, 64, 72, 80, 96, 100, 120, and 200.[23][24] Certain standard pitches such as 1/10 and 1/20 in inch measurements, which mesh with linear rack, are actually (linear) circular pitch values with units of "inches"[24]
When gear dimensions are in the metric system the pitch specification is generally in terms of module or modulus, which is effectively a length measurement across the pitch diameter. The term module is understood to mean the pitch diameter in millimeters divided by the number of teeth. When the module is based upon inch measurements, it is known as the English module to avoid confusion with the metric module. Module is a direct dimension, unlike diametral pitch, which is an inverse dimension ("threads per inch"). Thus, if the pitch diameter of a gear is 40 mm and the number of teeth 20, the module is 2, which means that there are 2 mm of pitch diameter for each tooth.[25] The preferred standard module values are 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0, 1.25, 1.5, 2.0, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40 and 50.[26]


File:Skupaj ogv q10ifps2fr6.ogv
Gear Cutting simulation (length 1m35s) faster, high bitrate version.
As of 2014, an estimated 80% of all gearing produced worldwide is produced by net shape molding. Molded gearing is usually eitherpowder metallurgy or plastic.[27] Many gears are done when they leave the mold (including injection molded plastic and die cast metal gears), but powdered metal gears require sintering and sand castings or investment castings require gear cutting or other machining to finish them. The most common form of gear cutting is hobbing, but gear shapingmilling, and broaching also exist. 3D printing as a production method is expanding rapidly. For metal gears in the transmissions of cars and trucks, the teeth are heat treated to make them hard and more wear resistant while leaving the core soft and tough. For large gears that are prone to warp, a quench press is used.


Overall gear geometry can be inspected and verified using various methods such as industrial CT scanningcoordinate-measuring machineswhite light scanner or laser scanning. Particularly useful for plastic gears, industrial CT scanning can inspect internal geometry and imperfections such as porosity.
Important dimensional variations of gears result from variations in the combinations of the dimensions of the tools used to manufacture them. An important parameter for meshing qualities such as backlash and noise generation is the variation of the actual contact point as the gear rotates, or the instantaneous pitch radius. Precision gears were frequently inspected by a method that produced a paper "gear tape" record showing variations with a resolution of .0001 inches as the gear was rotated.[24]
The American Gear Manufacturers Association was organized in 1916 to formulate quality standards for gear inspection to reduce noise from automotive timing gears;[28] in 1993 AGMA assumed leadership of the ISO committee governing international standards for gearing. The ANSI/AGMA 2000 A88 Gear Classification and Inspection Handbook specifies quality numbers from Q3 to Q15 to represent the accuracy of tooth geometry; the higher the number the better the tolerance.[29] Some dimensions can be measured to millionths of an inch in controlled-environment rooms.[29]

Gear model in modern physics[edit]

Modern physics adopted the gear model in different ways. In the nineteenth century, James Clerk Maxwell developed a model of electromagnetism in which magnetic field lines were rotating tubes of incompressible fluid. Maxwell used a gear wheel and called it an "idle wheel" to explain the electrical current as a rotation of particles in opposite directions to that of the rotating field lines.[30]
More recently, quantum physics uses "quantum gears" in their model. A group of gears can serve as a model for several different systems, such as an artificially constructed nanomechanical device or a group of ring molecules.[31]
The Three Wave Hypothesis compares the wave–particle duality to a bevel gear.[32]

Gear mechanism in natural world[edit]

Issus coleoptratus
While the gear mechanism was previously considered to be exclusively human-made, scientists from the University of Cambridge discovered that a common insect Issus, found in many European gardens, has in its juvenile form hind leg joints that form two 180-degree, helix-shaped strips with twelve fully interlocking spur type gear teeth. The joint rotates like mechanical gears and synchronizes Issus's legs when it jumps.[33][34]