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Lots, and Lots, of Dots

September 9, 2019

Many elementary school students dislike mathematics for the simple reason that it's never ending. After mastering integer fractions in third grade, they need to learn about decimal numbers in fourth grade. They find that a dot can be placed in a string of numbers to separate the integer part of a real number from its fractional part. This dot is commonly known as a decimal point, but there are other conventions for this decimal separator.

The decimal point is used as the decimal separator in about half the countries in the world, including the United States, the United Kingdom, the People's Republic of China, India, Japan, South Korea, Australia, and New Zealand. The other half uses a decimal comma instead. These comma countries include Russia, Germany, Norway, Sweden, Denmark, and The Netherlands (see figure). I've encountered scientific papers and reports that use each convention; and, while it's confusing at first, it becomes easy to translate commas to dots as you read.

Decimal separator map

A country map of decimal separators. Decimal point countries are shown in blue, decimal comma countries are shown in green, and countries using the Arabic momayyez (similar to a forward slash) are shown in red. No data were available for the countries marked in gray. (Illustration by User:NuclearVacuum, via Wikimedia Commons

It's unfortunate that such a small symbol as a dot is used for such an important concept. Early photocopiers were notorious for sprinkling their copies with random debris that might be taken for decimal points in some important numbers. Modern inkjet printers do the same in creation of their originals. A small splash of acid on a laboratory notebook page has the same effect. Often the only indicator is the presence or absence of a space between numbers.

The dot is also used as the mathematical symbol for the dot product of vectors. For vectors a and b with magnitudes, ||a|| and ||b||, and with an interior angle θ between them, this would be a·b = ||a|| ||b|| cosθ. The dot is also used as a multiplication symbol instead of x, since numbers are vectors on the number line with no interior angle, so the cosine is always one.

The dot is one of the binary characters of Morse code, the other being the dash. The dot indicates a short current pulse, while a dash indicates a longer pulse. Morse code was created by by Samuel F.B. Morse (1791-1872), inventor of the telegraph, and Alfred Vail (1807-1859). Vail had the important idea of taking an inventory of the movable type characters at a local newspaper to determine letter frequency as a means of creating an efficient code. Thus, the common letters "e" and "t" are represented by a single dot and a single dash.

Morse was also an artist, and not long after the 1944 creation of Morse code, other artists created oil paintings using dots of paint instead of strokes in a technique called pointillism. Pioneers in this technique were Georges Seurat (1859-1891) and Paul Signac (1863-1935). The best example of pointillism from this period is Seurat's, A Sunday Afternoon on the Island of La Grande Jatte (Un dimanche après-midi à l'Île de la Grande Jatte), a large oil painting of dimensions 81.7" x 121.25" that he completed in 1886 (see figure).

Georges Seurat: A Sunday Afternoon on the Island of La Grande Jatte

Georges Seurat's, A Sunday Afternoon on the Island of La Grande Jatte (Un dimanche après-midi à l'Île de la Grande Jatte).

(Located at the Art Institute of Chicago (Accession number 1926.224). Click for a larger image on Wikimedia Commons.)

Perhaps it was pointillism that inspired Leon Lenkoff of Louisville, Kentucky, to invent his "Magic pictures" in 1980.[1] His invention was a coloring book containing arrays of small, screen-printed dots of watercolor pigment within the outline images. Application of water with a brush or marker pen allowed coloring the images without the need for watercolor paints. Dots are also used in another type of children's activity book, connect-the-dots, that also trains number skill.

In Linux and other operating systems, hidden files or directories are prefixed with a dot. People familiar with the Linux command line interface use the dot command for executing a program, and many embedded systems utilize dot-matrix displays in which letters and numbers are formed by individual dots (see figure). Dot distribution maps, in which the density of dots shows the distribution of a variable. A famous example of this is the map of the 1854 Broad Street cholera outbreak in Soho, Westminster, that traces the source of the infection that killed more than 600 people to a particular water pump.

Example of a 5x7 dot-matrix display

Company name rendered in a 5x7 dot-matrix display. Embedded system displays are typically liquid crystal displays, but today's inexpensive and highly efficient Light-emitting diodes allow creation of more readable and aesthetic displays. (Created using Inkscape.)

Astronomy has its dots, also, in the form of the circled dot used as a symbol for the Sun, and Carl Sagan's iconic photo of Earth viewed at a distance as a pale blue dot. The Lewis electron dot structure was created in 1916 by American chemist, Gilbert N. Lewis (1875-1946), as an early explanation of the principles of chemical bonding. random dot stereograms were used as a tool in visual perception.

Next to a black hole, the singularity of which might be considered a dot, the most important dot in today's physics is the quantum dot. When you shrink a semiconductor to nanoscale dimension, to dot size, its optical and electronic properties are different from the bulk material. In particular, the optical and electronic properties of quantum dots change with size and shape, and ones with larger diameter will emit longer wavelengths of light, such as red, and ones with smaller diameters will emit shorter wavelengths, like blue. Such a property arises from size-dependent changes in the electron band energies. Two principal applications of quantum dots are as color-emitting elements in quantum dot displays; and the converse application, light-collecting elements in quantum dot solar cells.

A large international team of scientists from Beijing University of Chemical Technology (Beijing, China), Lawrence Berkeley National Laboratory (Berkeley, California), the University of California, Santa Cruz (Santa Cruz, California), the University of California, Berkeley (Berkeley, California), the University of Massachusetts Amherst (Amherst, Massachusetts), and Tohoku University (Sendai, Japan) has just published work on an unusual magnetic dot. These dots are droplets of an aqueous dispersion of magnetite (Fe3O4) nanoparticles formed by jet printing into oil, and they exhibit ferromagnetism, so they can be manipulated magnetically.[2-5]

When people think of magnets, they are thinking of ferromagnets, which are materials like iron that are attracted to a magnet or can be made into a magnet. Permanent magnetism arises when electron spins in the solid are locked into alignment by strong spin-spin interaction below a critical temperature called the Curie temperature. Another type of magnet is a paramagnet, a type of material that's attracted to a permanent magnet but can't be permanently magnetized. An example of this type of magnet is iron (II) oxide (ferrous oxide). Paramagnets have a weaker coupling between electron spins, so exposure to a magnetic field does not render them permanent magnets.

Ferrofluids are are colloid of nanoscale magnetic particles in a carrier fluid such as water or oil. To inhibit clumping, the particles are coated with a surfactant. While ferrofluids are attracted to a magnet, they are not ferromagnets and they can not be permanently magnetized. Mechanical engineers are more likely that other scientists and engineerss to have used ferrofluids, since they are used in rotary seals as ferrofluidic seals.[6] The aforementioned research team has discovered a way to couple the magnetic moments of superparamagnetic ferrofluidic nanoparticles of magnetite in a colloid to make a fluid magnet.[2]

Says research team leader, Tom Russell, a professor of polymer science and engineering at the University of Massachusetts, Amherst,
"We wondered, If a ferrofluid can become temporarily magnetic, what could we do to make it permanently magnetic, and behave like a solid magnet but still look and feel like a liquid?... We've made a new material that is both liquid and magnetic. No one has ever observed this before... This opens the door to a new area of science in magnetic soft matter."[4]

A magnetic droplet propelled through a solenoid coil

A magnetic droplet of magnetite (Fe3O4) nanoparticles propelled through a solenoid coil.

(Xubo Liu et al./Berkeley Lab image. Click for animated image.)

The researchers used a 3D-printing technique to print 1 mm droplets from a ferrofluid solution containing 20 nanometer magnetite nanoparticles.[4] They found that the nanoparticles formed a solid-like shell at the water-oil liquid-liquid interface, a phenomenon known as interfacial jamming, that produced a reversible paramagnetic-to-ferromagnetic transformation.[2,4] The droplets were magnetized by a solenoid coil, and their magnetic attraction caused them to clump together.[4] The 8 nanometer gap between nanoparticles at the jammed surface allowed magnetic alignment of the particles, and it was found that even smaller droplets than these, with diameters about the size of a human hair (about 75 micrometers), were also magnetic.[4]

Unlike solid magnets, these liquid can be reconfigured into different shapes while preserving their magnetic properties, including north-south dipole interactions.[2] Like permanent magnets, the ferromagnetic liquid droplets exhibit the properties of coercivity and remanence, and they're easy to control with external magnetic fields.[2] All the north-south poles in droplet arrays respond in unison, just as would the individual magnetic domains of solid ferromagnet.[4] The droplets can also be demagnetized.[4]

Magnetic droplets with different magnetic character

An array of magnetic droplet, 1 millimeter in diameter, with different magnetic character. The green droplets are paramagnetic without any jammed nanoparticles at the liquid interface, the red droplets are paramagnetic with nonmagnetic nanoparticles jammed at the interface, and the brown droplets are ferromagnetic with magnetic nanoparticles jammed at the interface.

(Xubo Liu et al./Berkeley Lab image.)

The small size of the droplets, combined with their unusual magnetic properties, suggest applications that range from artificial cells that deliver cancer therapies, to shape-changing flexible robots that adapt to their liquid environment.[4] Miniature robots can be devised to seek out diseased cells.[4] Says Xubo Liu, the study’s lead author who as a postdoctoral student discovered the droplets magnetism when he saw their movement on a magnetic stir plate, "What began as a curious observation ended up opening a new area of science... It's something all young researchers dream of, and I was lucky to have the chance... to make it a reality."[4]


  1. Leon G. Lenkoff, "Magic pictures," US Patent No. 4,212,393, July 15, 1980.
  2. Xubo Liu, Noah Kent, Alejandro Ceballos, Robert Streubel, Yufeng Jiang, Yu Chai, Paul Y. Kim, Joe Forth, Frances Hellman, Shaowei Shi, Dong Wang, Brett A. Helms, Paul D. Ashby, Peter Fischer, and Thomas P. Russell, "Reconfigurable ferromagnetic liquid droplets," Science, vol. 365, no. 6450 (July 19, 2019), pp. 264-267, DOI: 10.1126/science.aaw8719.
  3. Rémi Dreyfus, "An attractive, reshapable material," Science, vol. 365, no. 6450 (July 19, 2019), p. 219, DOI: 10.1126/science.aax8979.
  4. Theresa Duque, "New Laws of Attraction: Scientists Print Magnetic Liquid Droplets," Lawrence Berkeley National Laboratory Press Release No. 510-495-2418, July 18, 2019.
  5. Scientists Print Liquid Magnetic Droplets, Berkeley Laboratory YouTube Video by Marilyn Chung/Berkeley Lab, July 19, 2019 .
  6. Ronald E Rosensweig, "Magnetic fluid seals," US Patent No. 3,620,584, November 16, 1971.

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