# Draft:Distances

Distance (or farness) is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria (e.g. "two counties over"). In Draft:mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific set of rules, and provides a concrete way of describing what it means for elements of some space to be "close to" or "far away from" each other.

## Notations This is a foldable, or pliant, wood, double meter, or double-meter stick. Credit: Isabelle Grosjean ZA.

Def. a basic unit of length in the International System of Units (SI: Système International d'Unités) is called a metre, or meter.

Notation: let m represent a metre.

Notation: let km represent a kilometre.

## Measurements A measuring wheel or surveyor's wheel may be used for measuring lengths or distances around a kilometre. Credit: Suricata.

Def. a usually rigid flat, rectangular measuring or drawing device with graduations in units of measurement; a rule; a straightedge with markings; a measure is called a ruler.

Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the meter, second or degree Celsius.

For measuring distances around that of a kilometre, a measuring wheel or surveyor's wheel as in the second image down on the right may be used.

## Standards

Def. a principle or example or measure used for comparison is called a standard.

While a US penny may be rather precisely made and somewhat familiar, it is usually not considered a standard for length or size yet at slightly smaller scales suffices.

The public meter standard by Chalgrin in the second image down on the right is from the 18th century (36, rue de Vaugirard, 6th arrondissement of Paris, near the Luxembourg Palace). The public could bring meter rulers and check if they were accurate. A bar that just fit between the end stops was exactly one meter.

## Astronomical units

Notation: an astronomical unit is usually represented by au, or AU.

Def. an SI unit of length equal to 103 metres is called a kilometre, or kilometer.

Def. "a conventional unit of length equal to 149 597 870 700 m exactly" is called an astronomical unit (au).

Def. "9,460,730,472,580.8 km" is called the light-year (ly).

Def. a "vector quantity which denotes distance with a directional component" is called a displacement.

Radiation is usually thought of as projected from a source to a point or set of points at a distance from the source. Its displacement is the direction and distance from its source.

## Theoretical distances

Def. an amount of space between two points, usually geographical points, usually (but not necessarily) measured along a straight line is called a distance.

## Materials HRTEM lattice images and electron diffraction patterns (top-right insets) are taken along the  direction. Credit: Materialscientist. Image size is 1,750 x 810 pm. Credit: Boris J. Albers, Todd C. Schwendemann, Mehmet Z. Baykara, Nicolas Pilet, Marcus Liebmann, Eric I. Altman and Udo D. Schwarz. The symmetry of the protrusions in this image can only be explained by identifying them with the locations of the atoms in the lattice. Credit: Boris J. Albers, Todd C. Schwendemann, Mehmet Z. Baykara, Nicolas Pilet, Marcus Liebmann, Eric I. Altman and Udo D. Schwarz.

Def. a unit "of length; the thousandth part of one millimeter; the millionth part of a meter" is called a micron.

"As one spacecraft lurches and drags through the Earth's uneven gravity field, the second follows 210 km behind, measuring changes in their separation to the nearest micron (a thousandth of a millimetre)."

Def. an "SI subunit of length equal to 10-9 metres" is called a nanometre.

The second image down on the right is an integrated circuit die image of a STM32F103VGT6 ARM Cortex-M3 MCU (microcontroller) with 1 Mbyte Flash, 72 MHz CPU, motor control, USB and CAN. Die size is 5339x5188 µm. View is from a Scanning Electron Microscope looking at the 180 nanometre SRAM cells on the die.

The third image down on the right is a high-resolution, transmission electron micrograph (HRTEM) lattice image with an electron diffraction pattern (top-right inset) taken along the  direction; i.e., looking down the  axis. An image simulation is added in the bottom-left inset, and a model fragment of the crystal structure is on the right of the bottom-left inset. On the lower right of the composite image is a 1 nanometre (nm) marker.

Def. "10-12 of a metre" is called a picometre.

The fourth lower image down on the right has an image of 1,750 x 810 pm.

It is an atomic force microscope image of stacked graphene sheets in graphite.

This compares with the image on the left which was taken at a height of ~97 pm above the surface.

The image on the lowest right was at ~12 pm.

For most heights above the surface the features display a three-fold symmetry as on the right.

## Recent history

The recent history period dates from around 1,000 b2k to present.

Def.

1. a series of interconnected rings or links usually made of metal,
2. a series of interconnected links of known length, used as a measuring device,
3. a long measuring tape,
4. a unit of length equal to 22 yards. The length of a Gunter's surveying chain. The length of a cricket pitch. Equal to 20.12 metres. Equal to 4 rods. Equal to 100 links.,
5. a totally ordered set, especially a totally ordered subset of a poset,
6. iron links bolted to the side of a vessel to bold the dead-eyes connected with the shrouds; also, the channels, or
7. the warp threads of a web

is called a chain.

Def. a unit of length equal to 220 yards or exactly 201.168 meters, now only used in measuring distances in horse racing is called a furlong.

Def.

1. a trench cut in the soil, as when plowed in order to plant a crop or
2. any trench, channel, or groove, as in wood or metal

is called a furrow.

Def. the distance that a person can walk in one hour, commonly taken to be approximately three English miles (about five kilometers) is called a league.

## Mathematics

Def.

1. "a quantity that has both magnitude and direction"
2. the signed difference between two points or
3. an ordered tuple representing a directed quantity or the signed difference between two points

is called a vector.