Topography Topology

[Bradley Zweig]

Topographyis a branch of geography concerned with the natural and constructed features on the surface of land, such as mountains, lakes, roads, and buildings. Topology is a branch of mathematics concerned with the distortion of shapes. In this, the first of two articles, Ian Short explores topological problems using topography.

Suppose now that there are more than four blue and green towns. Can we still construct a dividing wall? Each town must be isolated (it must not touch any other town) and it must not lie on the coast. Try to build such a wall yourself using the example to the left, and experiment with other, different, configurations of towns. Can you make a conjecture about when a dividing wall exists?

This dividing wall begins at some arbitrary point on the boundary of the island, and works its way clockwise round the island, just slightly inland. Every so often the wall heads along a path inland, performs a quick loop round a blue town, and then heads back to the coast just next to the inland path. Once every blue town has been circled in this fashion the wall terminates at a point on the boundary. The same procedure can be applied with any configuration of towns, and thus we have proved Theorem 1.

(If you find this proof too informal, then see if you can do a better job yourself. Our whole discussion has been informal, so you may wish to work through the material again, fixing definitions and tightening reasoning.)

Is there a dividing wall for every collection of coastal and inland towns? No, there is not. Try to work out the simplest configuration of towns for which a dividing wall does not exist. You will probably come up with the example shown below, which we call the minimal configuration.

Why is there no dividing wall for this set of towns? Think about where a dividing wall would start and end. By symmetry of the configuration, we can assume that the wall starts in the north-west, between the western blue town and the northern green town. The wall finishes between two of the towns, either in the north-west, north-east, south-east, or south-west. In each case, the dividing wall fails to separate blue from green. Check the four cases yourself. This is the simplest configuration of towns for which there is no dividing wall because, for three towns, there is always a dividing wall, and for four towns configured in a different (topologically different) fashion, there is always a dividing wall.

Can we say precisely when a collection of towns has a dividing wall? Let's call a configuration of towns an alternating configuration if there are four particular coastal towns in the configuration whose colours are, in clockwise order, green blue green blue. In other words, an alternating configuration is one that contains the minimal configuration.

We have already seen that alternating configurations of towns do not have a dividing wall; for if an alternating configuration did have a dividing wall, then that wall would also be a dividing wall for the minimal configuration, which, as we have seen, is impossible. It remains to prove that each non-alternating configuration of towns has a dividing wall. This time we approach the problem as topologists, and our method is illustrated in the following figure:

We begin with the non-alternating configuration of towns shown in the top-left. You may have noticed that the shape of the island is unimportant for dividing walls problems: we can stretch and distort the shape of the island without affecting the existence of dividing walls. So, the first step is to deform our island to a circular shape.

Next, we can distort the interior and boundary of our disc, still without affecting the existence of dividing walls. So in the second step we push all the green towns to one half of the disc and we push the blue towns to the other half (as shown in the top-right of the figure above). It is easy to construct a dividing wall for the resulting collection of towns; we simply choose a diameter of the disc as a wall.

It remains only to unwind the deformations performed in steps one and two; this time we track how the diameter wall is distorted as we unwind. Finally, we end up with a dividing wall for our initial configuration. In this way we see that every non-alternating configuration of towns has a dividing wall, and this concludes the proof of Theorem 2.

History is littered with examples of dividing walls that have contributed to war and misery. Let us suppose that the people from our blue and green towns are more enlightened, although retain their animosity towards each other. Rather than building a wall, they decide to build two roads. One road connects all the blue towns and the other road connects all the green towns. The two roads are not allowed to intersect themselves or each other. The blue road starts at a blue town, and then passes through each of the other blue towns, finishing at another blue town. Likewise for the green road and the green towns. Let us call a pair of such roads connecting roads.

Our islands so far have pretty basic topography; just a few towns, walls, and roads. What if we add lakes, through which neither walls nor roads can be built? These lakes resemble holes in our shape. In the terminology of topology the number of holes is the connectivity of the shape. We also count the outside of the shape as a hole. This means that a disc has connectivity one, an annulus (a disc with a hole in the middle) has connectivity two, and the island on the left has connectivity five.

It seems that it is correct usage to talk about the topology of the brain, but not the topography of the brain. And it seems like correct usage to talk about the topography of human organs, but not the topology of human organs.

(1) is a branch of mathematics that studies of how properties of points and sets become connected spaces and how the properties of these spaces change or do not change under continuous transformations

When we speak about the topography of human organs (with careful respect to a certain anatomical surface, for example the thoracic cavity), we are talking about describing with a visual or verbal description of the location of features (organs) on that anatomical surface, inside of the body.

However, here's a tip if you ever find yourself without access to a dictionary, and you find a word you don't understand. A good starting point to help you guess the meanings of words in English of Latin/Greek origin, or at least get a rough idea of their meaning, is to learn the meanings of common suffixes.

There are many words that use these suffixes. To name but a few: geology, geography, theology, calligraphy, morphology, stratigraphy, meteorology, photography, biology, biography, criminology, radiology, radiography, etc.

Topography is a field of geoscience and planetary science and is concerned with local detail in general, including not only relief, but also natural, artificial, and cultural features such as roads, land boundaries, and buildings.[1] In the United States, topography often means specifically relief, even though the USGS topographic maps record not just elevation contours, but also roads, populated places, structures, land boundaries, and so on.[2]

Topography in a narrow sense involves the recording of relief or terrain, the three-dimensional quality of the surface, and the identification of specific landforms; this is also known as geomorphometry. In modern usage, this involves generation of elevation data in digital form (DEM). It is often considered to include the graphic representation of the landform on a map by a variety of cartographic relief depiction techniques, including contour lines, hypsometric tints, and relief shading.

Detailed military surveys in Britain (beginning in the late eighteenth century) were called Ordnance Surveys, and this term was used into the 20th century as generic for topographic surveys and maps.[5] The earliest scientific surveys in France were the Cassini maps after the family who produced them over four generations.[6] The term "topographic surveys" appears to be American in origin. The earliest detailed surveys in the United States were made by the "Topographical Bureau of the Army", formed during the War of 1812,[7] which became the Corps of Topographical Engineers in 1838.[8] After the work of national mapping was assumed by the U.S. Geological Survey in 1878, the term topographical remained as a general term for detailed surveys and mapping programs, and has been adopted by most other nations as standard.

An objective of topography is to determine the position of any feature or more generally any point in terms of both a horizontal coordinate system such as latitude, longitude, and altitude. Identifying (naming) features, and recognizing typical landform patterns are also part of the field.

A topographic study may be made for a variety of reasons: military planning and geological exploration have been primary motivators to start survey programs, but detailed information about terrain and surface features is essential for the planning and construction of any major civil engineering, public works, or reclamation projects.

Surveying helps determine accurately the terrestrial or three-dimensional space position of points and the distances and angles between them using leveling instruments such as theodolites, dumpy levels and clinometers.GPS and other global navigation satellite systems (GNSS) are also used.

Even though remote sensing has greatly sped up the process of gathering information, and has allowed greater accuracy control over long distances, the direct survey still provides the basic control points and framework for all topographic work, whether manual or GIS-based.

In areas where there has been an extensive direct survey and mapping program (most of Europe and the Continental U.S., for example), the compiled data forms the basis of basic digital elevation datasets such as USGS DEM data. This data must often be "cleaned" to eliminate discrepancies between surveys, but it still forms a valuable set of information for large-scale analysis.

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