Adapted From: A Guide to Key Science Concepts (CFGE, 2011)

What is the concept of scale?

The idea of scale refers to the naturally occurring range of magnitudes or sizes of such measurable properties as distance, time, density, and speed. Scientists study everything from the behavior of the elementary particles that make up all of matter, to galaxies that are unbelievably distant from the Earth. Thus, scientists in different disciplines work with scales that are unfamiliar to most people.

external image watch_vector_eps.png

Time and Distance Scales

For example, timescales that can be experienced directly by humans range from around a millisecond to a human lifetime. On the other hand, scientifically important timescales range from incredibly minute fractions of a second to billions of years.

Similarly, the distance scale used by scientists is much larger than the scale that is directly experienced by people: scientifically relevant distance scales range from less than a nanometer to twenty billion light years. One of the best ways to illustrate this is to simply list phenomena that occur from one extreme of a quantity to the opposite extreme. This has been done in Tables 1 and 2 for distance and in Tables 3 and 4 for time. Similar scales can be described for such scientific things as volume, density, viscosity, velocity, earthquake magnitude (the Richter scale), and other phenomena to which numbers can be attached.
external image CM-Handle.jpg

external image usa-2000-population-density.gif
Nonscientific examples of scale include the scales of population density (ranging from the average population density of most of South Dakota to the population density of cities like Tokyo) and debt (which, in the U. S., ranges from a penny’s worth of postage due, to the over 15 trillion dollars of the national debt). The idea of scale is also familiar to any student who has ever picked up The Guiness Book of World Records: this book’s purpose is to publicize the extreme magnitudes of measurable quantities (world’s tallest man, world’s shortest man, and so on).

Table 1
Range of Sizes of Small Objects
(Power of 10)
1 meter
Height of a child
1 centimeter
The radius of a penny; the length of a carpenter ant
1 millimeter
The diameter of a straight pin
1 micrometer
The diameter of some kinds of bacteria (sizes range from 0.1 micrometers to 10 micrometers.
1 nanometer
The diameter of the DNA molecule
1 angstrom
The diameter of the hydrogen atom
1 femtometer
The size of a proton

Table 2
Range of Sizes of Large Objects
(Power of 10)
10 meters
The depth of a middle-class suburban front lawn
1 hectometer
The length of a football field
1 kilometer
A short walk for a healthy adult
1 Megameter
The distance from San Diego to San Francisco or from Washington, DC to Boston, MA
1 gigameter
A round trip to the moon
1 terameter
The size of Jupiter's orbit around the Sun
10 petameters
One light year; a quarter of the distance to the nearest star
100,000 zetameters
The distance to the farthest observed galaxy
Table 3
Time Ranges (Less than a human lifetime)
Magnitude (Power of 10)
100 years
109 seconds
Human lifespan
10 years
108 seconds
Age of a fifth-grader
1 year
107 seconds
Time of human gestation
1 month
106 seconds
A lunar cycle
1 day
105 seconds
Lifespan of a mayfly
1 hour
103 seconds
The length of a TV show
1 minute
102 seconds
2 TV commercials
1 second
100 seconds
A heartbeat
0.1 second
10-1 seconds
An eyeblink
1 millisecond
10-3 seconds
A high speed camera exposure; a hummingbird wingbeat
1 microsecond
10-6 seconds
The time it takes light to cross a football field
1 nanosecond
10-9 seconds
The time it takes light to go one foot
10-24 seconds
10-24 seconds
Lifetime a the Z particle
10-35 seconds
10-35 seconds
The time that it took the universe to expand by a factor of 10100 during the expansion that followed the Big Bang
10-43 seconds
10-43 seconds
The earliest time in the universe that physics can explain; the time before which the concepts of time and space had no separate meaning.
Table 4
Time Ranges (Greater than a human lifetime)
Magnitude (Power of 10)
1000 years
1010 seconds
500 before America was discovered.
10,000 years
1011 seconds
The total length of human civilization.
1 million years
1013 seconds
The length of the existence of humans as a species
1 billion years
1016 seconds
One quarter the age of the Earth
10 to 20 billion years
1017 seconds
The approximate age of the universe
1035 years
1042 seconds
The predicted lifetime of the proton.
1073 years
1080 seconds
The predicted lifetime of a blackhole.
external image 675px-EM_Spectrum_Properties_edit.svg.png

Scales Describe Context

A second feature of the concept of scale is the use of the term to describe a context. The human scale, for example, is the scale of the world that humans directly perceive. Distances range from a fraction of a millimeter to the distance to the farthest point to which we normally travel; times range from milliseconds to around one hundred years; the wavelengths of light that we see goes from violet to red; and so on. The way that we think about reality is conditioned by our everyday experiences at our own scale. In fact, the scale that we perceive as normal changes over a lifetime. The world that a toddler sees is quite different from the world as seen by a six-foot tall adult man. Chairs that the man finds comfortable are formidable climbing challenges to the toddler; windows that a man sees the world through present only sky to a toddler; and so on.

Moving Away From the Human Scale

As we move farther from the familiar human scale, the world becomes less and less recognizable. The forces that operate and the phenomena that are important become vastly different. This can be illustrated by looking at the features of scales that are progressively more different from the human scale.

external image images?q=tbn:ANd9GcRHWZ4pHOXpfgyOj7GHsnUD_QQYwMhDFwlOSzHjzocClS3gORwDtw

The Effect of Gravity Changes with Scale

At the scale of insects, for example, the force of gravity becomes less important than it is at the human scale. Human beings are highly aware of the force of gravity. It holds us to the ground even in high winds; it makes the last months of pregnancy uncomfortable as a woman’s center of mass shifts into a position that the muscles of her back and abdomen do not support well; it makes the sport of weight-lifting possible. A human bodybuilder can hope to lift up to 500 pounds, about three times his own weight. Ants, on the other hand, can carry things up to 50 times their own weight.

What works at one scale may not work at another scale

Scaling effects vary depending on the phenomena observed. The giant insects in science fiction horror films are, sadly, not really possible for scale-dependent reasons. The ability of their scaled-up limbs to support weight would not counterbalance the effects of gravity: the strength of a support increases as a factor of its cross-sectional area, while the volume (and therefore the mass) of an object, when scaled up, increases as the cube. The legs of a giant grasshopper, for example, would collapse the first time it stood up. This problem, incidentally, also applies to the rest of its skeleton-the materials used and the design of the exoskeleton would have to be radically changed to support the giant grasshopper’s bulk.

external image 7509323-a-surreal-photo-manipulation-portraying-giant-insects-and-spiders-on-an-arizona-highway-near-an-actu.jpg

Scale changes can be large or small

The change in scale represented by the fictional giant grasshoppers is relatively small: real grasshoppers are a few centimeters long, while Hollywood’s giant grasshoppers were perhaps 10 to 100 meters long. The factor by which they were scaled up was at most 10,000X. This isn’t much when it’s compared to the range of distances found in nature, as a quick reference to Tables 1 and 2 will show.

Microscopic Scale

At scales more different from our own, scale effects become even more pronounced. At the scale of bacteria (a factor of about 1,000,000X smaller than we are), for example, the motions of individual molecules are important, so important that they can be readily detected with a microscope. The wiggling of microscopic specimens suspended in water is known as Brownian motion. It is a direct result of the random thermal motion of molecules surrounding the specimen. At temperatures above absolute zero, all atoms and molecules are moving in a number of different ways: they rotate, they vibrate, and they move from place to place (they translate). Every molecule in a liquid, for example, is moving around in random directions. If lots of molecules bang into a bacterium from one side at the same time that only a few bang into it from the other side, the bacterium will move a small distance-but a distance that is readily apparent to the observer using the microscope. Needless to say, the movement of individual molecules is not nearly as important at the human scale as it is at the bacterial scale.

At the bacterial scale, other unusual effects can be seen as well. Gravity, for example, is much less important to a bacterium than it is to us. If a flask containing a culture of E. coli in nutrient broth is left in the refrigerator for a day so that the bacteria are too cold to swim, they will gradually settle to the bottom, a process that takes hours. It certainly wouldn’t take a person hours to fall a distance of a few inches in water. In addition, the water that the bacterium experiences have different important properties than it does for us. The water “feels” more viscous to the bacterium. Water molecules in a liquid are connected to each other by hydrogen bonds, which are relatively weak and break and reform constantly as the water molecules move thermally. To move, a bacterium must push against this interlocked network of hydrogen-bonded water molecules, a process that is more difficult for it than for us because of the relatively smaller force that it can exert. In fact, the surface of the water represents an impenetrable barrier for the bacterium, as the force that it can generate by spinning its flagellae is much smaller than that needed to break the water’s surface tension (which is itself a direct consequence of the hydrogen bonding between water molecules in the liquid).

Atomic Scale

If one descends further, to the scale of the atom, conditions become even more different from those prevailing at our scale. At the nanoscale, the laws of quantum mechanics become important. Instead of being smooth, things are quantized: for example, an electron moving in an atom can have only a few permitted energies, and all other energies are forbidden to it. If this were the case at our level, a car, say, could go five miles an hour or fifty miles an hour, but couldn’t go, say, forty miles an hour; and the transition between five and fifty miles an hour would occur instantaneously, with no intermediate speeds. In addition, the uncertainty principle becomes important: you can’t know both the position of a particle and its speed and direction of motion at the same time. Another strangeness is that the world at this level is mostly empty space: electrons can be best thought of as a buzzing fog around the atomic nucleus, and the size of the nucleus relative to the size of the atom can be compared to that of a ping-pong ball in the Astrodome. Advances in the field of nanotechnology take advantage of differences that occur at very small sizes.
external image images?q=tbn:ANd9GcQ5y5ZQvAA7BBkkr_l1jYAAVjRSDmigfRpabEeV2se1FZQsSk_xJA

external image images?q=tbn:ANd9GcSHDUfzVkYokPl34xzsvjmlmfA3fIzyPvbjseML4OLCEpPaaFgB

Large Scales

Clearly, our intuition and our perception of reality fail us at small distances. They also fail us at the largest distance scales. Consider the fundamental theorem of plane geometry: parallel straight lines will never intersect. Here, “straight line” means that the line connects any two points on it by the shortest distance between them; “parallel” means that a line perpendicular to one will also be perpendicular to the other. This fundamental theorem seems perfectly obvious, natural, and hardly worth stating (except by mathematicians). On large distance scales, however, it is wrong: parallel lines meet at some astronomically large scale. Our three-dimensional space is curved, although the curvature is so slight that one needs to examine billions of light years of distance to detect it.

To illustrate this point, one can consider a small ant on the surface of a large sphere (see figure at right). She would believe that the surface was flat (just as, for most of human history, people have thought that the Earth is flat.) Suppose this ant explored two lines on the sphere, corresponding to the north-south lines given by 120 degrees longitude and 100 degrees longitude, as illustrated in Figure 1. These two lines always connect any two points by the shortest distance between them, and they have the same perpendicular (the equator); they are thus two parallel straight lines. Yet they intersect at the poles. Similarly, in our universe, it is believed that two parallel straight lines will eventually intersect. In fact, if you travelled in a perfectly straight line, without turning around, you would eventually reach your starting point!

external image images?q=tbn:ANd9GcRUqEbHThchyRFNkghhSdyYTn13Z62xizG3XZhIDbpLVfXvNnNX
As illustrated by the preceding examples, the world is very different at different scales, and at the most extreme scales, normal human intuition can be entirely wrong. The concept of scale and an appreciation of the differences between events occurring at different scales is important for an understanding of science. Scientists working at different scales gradually develop an intuitive feeling for the scale context of their work. A quantum mechanic and an auto mechanic will thus have different, but in a way equally valid, appreciations of the world.

Rationale for Teaching the Concept

One of the most important features of science is that it helps us to better define our place in the universe. A proper appreciation of the concept of scale and of the differences that ensue with a change of scale will expand a child’s understanding of where she fits in and of the vast array of other possibilities that exist.

Suggested Applications:

The concept of scale can be illustrated in a variety of ways. Each of the applications suggested below could be adapted for use at most levels. In addition to the applications suggested here, the concept of scale can be taught indirectly when a scientific phenomenon occurs at a scale different from our own is studied. When microorganisms are studied, for example, the sizes and features of the microscopic world can be compared with the features and sizes to which humans are accustomed.

  1. Have students make lists similar to those in the tables above for a variety of different phenomena. Encourage students to come up with the numbers themselves through research and measurements of their own. For example, you could ask students to come up with a distance scale that illustrates the range of distances that they have experienced and illustrate it with examples.
  2. Have students physically model different scales. For example, a timeline is a good way to model the geologic timescale. If it is done reasonably accurately, it gives students a good feeling for how long the earth has been around and how short a period of time humans have been around in comparison. The scale of the solar system can be modeled by students. Give them a pea and tell them that it represents the Earth. Then ask them to tell you how they would represent the other planets, the moon, and the sun and where they would place these objects relative to the pea. These physical demonstrations give students an intuitive feeling for the magnitudes of scientific phenomena, particularly if they have to work out the features of the physical model themselves.
  3. Have students look at the effects of small changes in scale. Have them observe the animals directly to determine the kinds of differences that exist between the human scale and the animal's. Ask students to imagine how the world would look from the point of view of the animal. Have them draw pictures of the world as seen from the animal’s point of view, write stories about how it would feel to be the size of the animal, and so on.
  4. Have students do some engineering to look at scale effects (strength of supports, types of materials employed, and so on). A visit to a building site or two would amplify their understanding of the different materials required to make buildings on different scales (the materials for a house are different from those needed for a skyscraper, for example.) An architect or engineer could also be brought in to talk about scale effects and design.
  5. Have students design their own system of units and construct scales using them. This deepens their understanding of the use of units and of the human scale at the same time.

To illustrate scale, see the following sources:

Eames, R., & Eames, C. (2010). Powers of Ten [webpage with video]. Retrieved from

Parry-Hill, M. J., Burdett, C. A., & Davidson, M. W. (2009). Secret worlds: The universe within [web-based simulation]. Retrieved from