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 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
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.

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

Radiation
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.

Measurements
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.

Depths
Def. "the [vertical distance] height below a surface; the [degree to which] amount that something is deep" is called a depth.

"In soil, estimates are that 80 to 99% of the microorganisms remain unidentified (1)."

"The soil at the Arlington site is a Plano silt-loam. The 20-cm-deep A horizon is a silt-loam and contains 4.4% organic matter. The loess mantel is >1.25 m deep. Four 2.5-cm-diameter soil cores were taken from the top 10 cm of a clover-grass pasture at the Arlington Agricultural Research Station. The soil samples were immediately placed on dry ice, mixed, and then stored at -70°C prior to DNA extraction. Soil analysis was done by the Soil Testing Laboratory of the University of Wisconsin—Madison as described by Schulte et al. (40). The soil sample contained 13% sand, 70% silt, 17% clay, 4.4% organic matter, 0.3% total N, 400 ppm of K+, and 98 ppm of P. The soil pH was 6.5. The site is well drained, with groundwater more than 25 m below the surface. Two-thirds of the 79-cm annual rainfall occurs from April to October. The site has an average of 165 frost-free days."

In the image on the right, A, B, and C represent the soil profile: A is the topsoil; B is a regolith; C is a saprolite, a less-weathered regolith; the bottom-most layer represents the bedrock.

Stone spheres
Def. the "distance across a circle in [length of any] a straight line [between two points on the circumference of a circle that passes through the centre/center] crossing the midpoint [of the circle] " or the "maximum distance between any two points in a metric space" is called a diameter.

In June 2014, the Precolumbian Chiefdom Settlements with Stone Spheres of the Diquis was added to the UNESCO list of World Heritage Sites.

The spheres range in size from a few centimetres to over 2 m in diameter, and weigh up to 15 tons. Most are sculpted from gabbro.

The culture of the people who made them disappeared after the Spanish conquest.

The first scientific investigation of the spheres was undertaken shortly after their discovery and published in 1943 in American Antiquity, attracting the attention of Samuel Kirkland Lothrop of the Peabody Museum of Archaeology and Ethnology at Harvard University. In 1948, he and his wife attempted to excavate an unrelated archaeological site in the northern region of Costa Rica. In San José he met Doris Stone, who directed the group toward the Diquís Delta region in the southwest ("Valle de Diquís" refers to the valley of the lower Río Grande de Térraba, including the Osa Canton towns of Puerto Cortés, Palmar Norte, and Sierpe.

Surface areas
Def. a "measure of squared distance [the extent of a surface] ; it is measured in square units" is called an area.

In the image on the right, dense forest surrounds the city center of this Classic-era Maya site (top) Tikal. Laser mapping of the same view (bottom) reveals structures and causeways hidden by the jungle.

"Lidar (a type of airborne laser scanning) provides a powerful technique for three-dimensional mapping of topographic features."

"Lowland Maya civilization flourished from 1000 BCE to 1500 CE in and around the Yucatan Peninsula."

"In 2016, the Pacunam Lidar Initiative (PLI) undertook the largest lidar survey to date of the Maya region, mapping 2144 km2 of the Maya Biosphere Reserve in Guatemala."

"Analysis identified 61,480 ancient structures in the survey region, resulting in a density of 29 structures/km2. Controlling for a number of complex variables, we estimate an average density of ~80 to 120 persons/km2 at the height of the Late Classic period (650 to 800 CE). Extrapolation of this settlement density to the entire 95,000 km2 of the central lowlands produces a population range of 7 million to 11 million."

"Settlement distribution is not homogeneous, however; we found evidence of (i) rural areas with low overall density, (ii) periurban zones with small urban centers and dispersed populations, and (iii) urban zones where a single, large city integrated a wider population."

"The PLI survey revealed a landscape heavily modified for intensive agriculture, necessary to sustain populations on this scale. Lidar shows field systems in the low-lying wetlands and terraces in the upland areas. The scale of wetland systems and their association with dense populations suggest centralized planning, whereas upland terraces cluster around residences, implying local management. Analysis identified 362 km2 of deliberately modified agricultural terrain and another 952 km2 of unmodified uplands for potential swidden use. Approximately 106 km of causeways within and between sites constitute evidence of inter- and intracommunity connectivity. In contrast, sizable defensive features point to societal disconnection and large-scale conflict."

Ranges
Def. the "distance from a person or sensor to an object, target, emanation, or event", the maximum "distance of capability (of a weapon, radio, detector, fuel supply, etc.)", or the "set of values (points) which a function can obtain" is called a range.

At right is an "eerie view of Jupiter's moon Io in eclipse ... acquired by NASA's Galileo spacecraft while the moon was in Jupiter's shadow. Gases above the satellite's surface produced a ghostly glow that could be seen at visible wavelengths (red, green, and violet). The vivid colors, caused by collisions between Io's atmospheric gases and energetic charged particles trapped in Jupiter's magnetic field, had not previously been observed. The green and red emissions are probably produced by mechanisms similar to those in Earth's polar regions that produce the aurora, or northern and southern lights. Bright blue glows mark the sites of dense plumes of volcanic vapor, and may be places where Io is electrically connected to Jupiter."

"North is to the top of the picture, and Jupiter is towards the right. The resolution is 13.5 kilometers (8 miles) per picture element. The images were taken on May 31, 1998 at a range of 1.3 million kilometers (800,000 miles) by Galileo's onboard solid state imaging camera system during the spacecraft's 15th orbit of Jupiter."

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).

Hubble constants
Def. "a number that expresses the rate of expansion of the universe; the velocity at which a distant object is receding from the Earth divided by its distance; its reciprocal gives the age of the universe" is called the Hubble constant.

"Measurements of the rate of cosmic expansion using different methods keep turning up disagreeing results."

Cepheid variables are "a class of stars that flicker regularly [where using] this pulsation rate, [allows the calculation of] the universe's expansion rate, but that number doesn't match with values derived from other cosmic phenomena, such as the echo of the Big Bang known as the cosmic microwave background radiation."

"Using data from the European Space Agency's (ESA) Planck satellite, [...] the rate [is estimated] to be 46,200 mph per million light-years (or, using cosmologists' units, 67.4 kilometers/second per megaparsec)."

"But calculations using pulsating stars called Cepheids suggest it is 50,400 mph per million light-years (73.4 km/s/Mpc)."

"If the first number is right, it means scientists have been measuring distances to faraway objects in the universe wrong for many decades. But if the second is correct, then researchers might have to accept the existence of exotic, new physics."

"Back in 1929, [Edwin Hubble] noticed that more-distant galaxies were moving away from Earth faster than their closer-in counterparts, [producing] a linear relationship between the distance an object was from our planet and the speed at which it was receding."

"That means something spooky is going on. Why would we be the center of the universe? The answer, which is not intuitive, is that [distant objects are] not moving. There's more and more space being created between everything."

"Things got weirder in the late 1990s, when two teams of astronomers noticed that distant supernovas were dimmer, and therefore farther away, than expected. This indicated that not only was the universe expanding, but it was also accelerating in its expansion."

"For the last 10 years, the Planck satellite has been measuring the cosmic microwave background, a distant echo of the Big Bang, which provides a snapshot of the infant universe 13 billion years ago. Using the observatory's data, cosmologists could ascertain a number for the Hubble constant with an extraordinarily small degree of uncertainty."

"It's beautiful. But, it contradicts what people have been doing for the last 30 years."

"But estimates of the Hubble constant using Cepheids don't match the one from Planck. The discrepancy might look fairly small, but each data point is quite precise and there is no overlap between their uncertainties. The differing sides have pointed fingers at one another, saying that their opponents have included errors throwing off their results."

"If the Cepheids teams are wrong, that means astronomers have been measuring distances in the universe incorrectly this whole time. But if Planck is wrong, then it's possible that new and exotic physics would have to be introduced into cosmologists' models of the universe. These models include different dials, such as the number of types of subatomic particles known as neutrinos in existence, and they are used to interpret the satellite's data of the cosmic microwave background. To reconcile the Planck value for the Hubble constant with existing models, some of the dials would have to be tweaked, but most physicists aren’t quite willing to do so yet."

Materials
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 [0001] direction; i.e., looking down the [0001] 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.