Floating Point Operation
Cartographs by Katherine E. Bash
Text by William L. Fox
Bonneville Salt Flats, Utah
Research Base: Center for Land Use Interpretation, Wendover, Utah
A floating-point number is one in which the position of the point is determined within the number itself. The term floating point indicates that there is no fixed number of digits before and after the decimal point, and therefore the decimal point is unanchored. A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number.
Floating-point calculations involve approximation because the result of an operation might not be exactly representable. Floating-point numbers are of limited precision and can therefore only represent a finite set of values, but allow a large range of magnitudes to be represented within a given size of field, which is not possible in fixed-point notation. Floating-point representations often include the special values +∞ or -∞ (positive and negative infinity).
The four cardinal directions of north, east, south, and west are the basis for the worldwide graticule, or rectilinear grid that is used to determine a location relative to the surface of the Earth via latitude and longitude. The mapping of the planet’s spherical surface onto a two-dimensional plane can only be approximate as curved surfaces have two finite radii of curvature, while maps, like all plane surfaces, have two infinite radii of curvature.
In response to these constraints, cartography uses various map projections—graphic systems that attempt to transfer positions on the planet’s spheroidal surface to corresponding points on a sheet of paper—to diminish the distortion. The number of points on a sphere is infinite, and the process of transformation remains approximate. It is a floating-point world.
• in·fi·nite Late Middle English : from Latin infinitus, from in- ‘not’ + finitus ‘finished, finite’ (see finite ). adj 1: limitless or endless in space, extent, or size; impossible to measure or calculate. 2: very great in amount or degree. 3: Mathematics greater than any assignable quantity or countable number. 4: Mathematics (of a series) able to be continued indefinitely. 2 Grammar another term for nonfinite. noun (the infinite): a space or quantity that is infinite.
Boxing the Compass
“Boxing the compass” is the term for naming the 32 points on a compass rose, and has long been a basic navigational skill required of all sailors. The compass rose, as a set of calibrated rings, still appears on many Global Positioning Satellite (GPS) devices.
Positioning oneself in isotropic spaces relative to the wind is a skill often deployed by native peoples and experienced explorers when landmarks become invisible. In order to do this it is necessary to know from what direction the wind is coming, which is often possible to determine only by knowing the time of year, the time of day, and whether or not storm systems are entering or leaving the area.
Because wind shapes features on the ground, one can determine the direction of various winds by looking at the terrain. On the Bonneville Salt Flats, for example, small ridges on its hardened surface may run north and south, or at right angles to the prevailing westerlies. When the flat is underwater, salt stalactites grow along the direction the water is being pushed by the wind.
Dead Reckoning (backsight)
At the far end of the southernmost extension of the Bonneville Dike, where it meets I-80 and then bends eastward, Floating Island is apparent as an outlier of the Silver Island Mountains, connected under the basin fill. Floating Island, material from which has been drawn out on the playa for almost 40 kilometers, is a landmark stretched into a landline, thus sharing the characteristics with the interstate freeway, railroad tracks, and military fences that create discrete and bounded spatial units within the larger area.
At the same time, these lines drawn on the land have a temporal limit and are subject to geomorphologic forces, principally erosion, and geologic ones such as isostasy. Even the horizon proves variable, depending on the elevation of the viewer.
The detritus on the playa at this elbow is literally (littered) beach wrack, which is itself used to construct temporary monuments awaiting the arrival of the seasonal runoff.
To the observer standing on the playa, Floating Island is a monolithic and symmetrical landmark, an indexical reference point in the landscape. As the observer begins to move around the island, however, it fractures into a series of planes and elevations. The island turns out to have several summits, not one; and, when circumnavigating its shoreline, what from afar seemed simply circular reveals its fractal nature.
Just as movement through space begins to make apparent the geomorphological complexity of the landscape, so does movement over time. The crust of Earth, which is hard over human-scale periods of time, acts like a viscous fluid over geologic durations. The crust is very thin in the Great Basin and being stretched by tectonic forces, the distance between Salt Lake City and Reno increasing up to one meter per century. As a result, even the supposedly stable elevation of the ancient benches is complicated by a high degree of isostatic crustal rebound.
Looking at the Silver Island Mountains from atop Floating Island, a lighter band of material is apparent at their base. These alluvial fans reside below the Bonneville Bench, which appears as a level shoreline in the landscape, but in fact is subtly inclined. 30,000 years ago Lake Bonneville had a volume of more than 10,000 cubic kilometers of water. The water load of the Pleistocene lake caused hot rock in this thin local crust to flow outward several hundred kilometers deep in Earth. This allowed the weight of the lake to depress the crust by 70 meters at its deepest point. Once the lake waters began to decline, the crust started to rebound, a process that continues today.
Although Floating Island is more than 100 kilometers west of what was the lake’s deepest point, its shorelines were somewhat depressed and have since risen as the water evaporated. The salt flats exist on the playa because of a low topographical divide separating this playa from the Great Salt Lake—a divide created by isostatic rebound, by movement in the landscape.
The panorama has been used for centuries as a way of capturing both the environment and one’s position within it, the vantage point of the observer invisible yet implicit, determined by rotation around around a vertical axis. Mirrors, on the other hand, are used to provide us with a reflection of the observer, or whatever is mirrored. The Bonneville Salt Flat is a nearly isotropic space, the only relief from homogeneity in all directions the silhouettes of distant mountains. This is view with zero-point perspective, as there are no lines to converge in the distance, and a mirror does not change that fact.
The salt flats are intermittently submerged beneath a body of water several square kilometers in extent but only a few inches deep, a lake that the daily winds move unpredictably around the playa. The interaction of wind and water, as well as the severe temperature inversions create multiple reflections, many of which are inferior mirages, those caused when the conjunction of colder air atop warmer air heated by the ground create a mirror line that projects the sky.
When walking out a kilometer into the water from the end of the Bonneville Salt Flats access road and toward Floating Island, the landmark appears doubled in the water, but also doubled by mirage, the mirage then doubled in the water. This makes the view the same in all directions, above and below as well as around the observer, this increasing the isotropy.
A mirror is defined as a reflective surface that reflects a clear image, and is a word from Old French, mirour, that means “look at.” Agnosia is a word coined in German from the Greek for “not knowing.” Mirror agnosia, a neurological condition also known as “looking glass syndrome,” describes a spatial confusion in which a person mistakes the location of a reflected object as actually being in or behind the mirror. They do not know what they are looking at and will often try to reach through the mirror as if it were a window.
Floating Island appears symmetrical and monolithic enough from many vantage points such that, when viewing one side of it reflected by a mirror, it is difficult to perceive that one is not simply looking through clear glass at the other side, but is in fact seeing a reflection. This illusion is compounded by the fact that mirrors do not reverse their reflections. If an observer stands in front of a mirror and raises the left hand, the observer will not see the right hand raising, but the left. In essence, the mirror has convinced the observer that reality has been switched from front to back, not left to right.
It is logical in those terms for a person to think an object reflected in a mirror would reside behind the reflecting surface, in a territory that is, in fact, a region replaced by the mirror and lost to inspection by the imposition of a reflection.
As soon as one drives south from Floating Island up onto the dike, a fence becomes visible to the east, a line that slowly converges upon the observer’s route until, within 3 kilometers south of the island and less than 30 meters from the dike, it makes an abrupt and almost right-angle turn to the east. The fence is the largest artefact along the causeway, and this angle marks the southwestern corner of the northern portion of the Utah Test and Training Range (UTTR).
The UTTR consists of two parcels, the other being located south and east of Wendover. Together they form the largest combined restricted land and airspace in North America. The 1570-square-kilometer northern area is used primarily for munitions testing, including bombs with up to a ten-kiloton yield, the largest conventional weapons tested in North America outside of the Nevada Test Site. A small sign at the corner warns that this is a “controlled area” and trespassers may be detained, searched, and prosecuted.
When standing at the corner and peering down the fence lines, one cannot see their ends; the poles become smaller and smaller until they appear to drop below the horizon, or fade into a mirage. Driving farther south, the fence becomes entirely invisible within 3 kilometers. It becomes obvious that, when looking down the fence what makes it disappear is not the curvature of the Earth, but the inability of our eyesight to distinguish individual objects at that distance.
Floating Point Operation
Texts by William L. Fox.
Center for Land Use Interpretation, Wendover, UT