Beyond Ultra Large and Extra Small

Metric (SI) prefixes come in handy for expressing both large and small quantities. So we have gigawatt power systems and terabyte hard drives at the high end, and we have nanoscale electronics and femtosecond lasers at the low end.

Advances in scientific research and technology keep pushing us to new frontiers in the domain of metric prefixes.

In a 1998 article in Scientific American, Philip Morrison and Phylis Morrison described the escalation in the amount of stored knowledge since the days of the famed library in ancient Alexandria. The library probably shelved the equivalent of 50,000 books today. Counting the literature of China, India, Iraq, and Iran, the total number of distinct books available 2,000 years ago might have been about 100,000.

In 1945, a librarian at Harvard estimated that "worthwhile" printed material amounted to about 10 million books. With one byte as a unit of information representing roughly the equivalent of a single character of English text, it's possible to store that text as 10 trillion bytes, or 10 terabytes, of data. Images, drawings, photographs, audio and video recordings, and huge databases, however, require much larger amounts of storage space.

The U.S. Library of Congress now holds more than 34.5 million books and other print materials. Because it also has 3.3 million sound recordings, 13.4 million photographs, 5.4 million maps, and other byte-hungry material, however, the total store comes to a couple of petabytes (a million billion bytes).

One can plausibly estimate that fresh text, including newspapers, amounts to less than 100 terabytes annually. Recorded music, films, photographs (including family snapshots), and other materials add considerably to that accumulation.

"The biggest byte makers are the television stations of the world," the Morrisons wrote. "Although it is hard to correct for innumerable repeats, our best source puts their originality at one tenth of all they send out and so allots them under 100 petabytes annually."

To assess the totality of information production, one can also try to include ephemeral signals conveyed from one person to another. The sounds of telephone calls add up to some 1,000 petabytes worldwide, or a few exabytes. Face-to-face speech provides several more exabytes of data.

At this stage, the summing gets trickier. The Morrisons simply concluded, "No estimate of the eventual human store seems quite credible as yet."

In a 2011 online article in Science Express, Martin Hilbert and Priscila Lopez provided new estimates of the world’s technological capacity to store, communicate, and compute information. They estimated that in 2007 humankind was able to store 2.9 x 10²⁰ optimally compressed bytes, communicate nearly 2 x 10²¹ bytes, and carry out 6.4 x 10¹⁸ instructions per second on general-purpose computers.

These estimates put us in a realm of metric prefixes that relatively few people yet know. In 1991 as part of the International System of Units (SI), the General Conference on Weights and Measures (Conférence Générale des Poids et Mesures) adopted new prefixes representing 10²¹, 10²⁴, 10^-21, and 10^-24. This means that 1,000 exabytes equals 1 zettabyte (ZB), and 1,000 zettabytes equals 1 yottabyte (YB).

I first encountered the new prefixes in 1993 when researchers measured voltages in a superconducting circuit so small that they had to use the term milliattovolt, where "atto" stands for 10^-18. The proper term is zeptovolt.

Here's the current table of official metric (SI) prefixes:

Power of 10	Prefix	Symbol	Power of 10	Prefix	Symbol
24	yotta	Y	-1	deci	d
21	zetta	Z	-2	centi	c
18	exa	E	-3	milli	m
15	peta	P	-6	micro	m
12	tera	T	-9	nano	n
9	giga	G	-12	pico	p
6	mega	M	-15	femto	f
3	kilo	k	-18	atto	a
2	hecto	h	-21	zepto	z
1	deka	da	-24	yocto	y

Many of the prefixes come from Greek and Latin words, often via French. "Zepto" is derived from the Latin septem, meaning 7, because this is the seventh prefix in the system of metric prefixes. The s was replaced by z to avoid confusion with the abbreviation for the second. The prefix "zetta" was coined to parallel "zepto." Similarly, "yocto" is derived from the Latin octo, meaning 8, and "yotta" parallels that term.

You can find a guide to units of measurements and their history at A Dictionary of Units of Measurement.

You might be interested to know that an attoparsec is a distance of about one inch (3.1 centimeters). The distance for Earth of the most remote object yet observed in the universe is about 125 yottameters (13.2 billion light-years). The diameter of the largest known galaxy is about 53 zettameters. At the other end, an atomic mass unit equals 1.66 yoctograms.

The Convert Auto page (a student project) offers a handy tool for converting from one unit of measurement to another in a wide range of fields.

Now you can really start talking ultra large and extra small.

Originally posted July 27, 1998

References:

Hilbert, M., and P. Lopez. The world’s technological capacity to store, communicate, and compute information. Science Express (Feb. 10, 2011).

Morrison, P., and P. Morrison. 1998. Wonders: The sum of human knowledge? Scientific American 279(July):115.

Peterson, I. 1993. Measuring superconductor magnetic noise. Science News 143(Jan. 16):37.

Fermat's Natural Spirals

A typical daisy is a star-like flower. It features a fringe of white or colored petals and a central disk of tubular florets. Each floret is itself a tiny flower.

This daisy has 21 white petals and a yellow central disk of tubular florets. Photo by Kenneth Peterson.

The tightly packed florets at a daisy's center have an intriguing arrangement. The florets get larger at greater distances from the center. And there are hints of clockwise and anticlockwise spirals in the pattern.

A close-up view of a daisy's disk of florets reveals intriguing spiral patterns. Photo by Kenneth Peterson.

One way to model such a pattern is to start with a curve called Fermat's spiral. This curve is also known as a parabolic spiral. It's given by the polar equation

r = k a^1/2

where r is the distance from the origin, k is a constant that determines how tightly wound the spiral is, and a is the polar angle.

This type of spiral has the property of enclosing equal areas with every turn.

Fermat's spiral.

By placing points (disks or polygons) centered at regular angular intervals along such a spiral, you can create a variety of intriguing patterns—depending on the angle you choose to use. Using the angle 222.49 degrees (a value related to the golden ratio, 1.618034. . . ), you get a pattern with an even packing of polygons (or disks). It closely resembles a daisy's florets.

By placing points at regular angular intervals along Fermat's spiral, you get a pattern resembling that of a daisy's central florets. Courtesy of Robert Krawczyk.

By choosing other angles, you get intriguing variants. Each choice gives a different pattern of secondary spirals, some winding clockwise and others anticlockwise, which form an interlocking system. Robert Dixon explores some of these possibilities in his book Mathographics.

Using larger numbers of points and smaller angles produces patterns with a variety of secondary spirals and, often, with radial lines that become evident toward the edges. Michael Naylor has investigated a variety of such patterns (see "Golden Blossoms, Pi Flowers").

Robert J. Krawczyk has taken the generation of these patterns another step further, creating striking images of eerie ripple patterns. He calls these circular designs "Fermat's spiral mandalas."

Krawczyk starts by combining several spirals to create one complex pattern.

Four different individual spirals (top) can be combined in various ways to create new, complex patterns (bottom). Courtesy of Robert Krawczyk.

By placing points at fixed angular intervals along these curves, he gets very elaborate patterns that show a variety of features.

An example of a Fermat's spiral mandala. Courtesy of Robert Krawczyk.

To finish his images, Krawczyk enhances the texture and gives them a coppery glow. Examples of such mandalas can be seen at his website.

Courtesy of Robert Krawczyk.

A mandala's complex circular design, with its symmetrical and radial balance, is intended to draw the eye to its center, Krawczyk says. Fermat's spiral, in particular, "is a natural basis for this inward draw."

For those who prefer a more natural look, all it takes is a close view of a daisy's central disk.

Photo by Kenneth Peterson.

Originally posted September 5, 2005.

References:

Dixon, R. 1991. Mathographics. Dover.

Krawczyk, R.J. 2005. Fermat's spiral mandalas.

Naylor, M. 2002. Golden, √2, and π flowers: A spiral story. Mathematics Magazine 75(June):163-172.