Roman Urbanism
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The Romans privileged an orthogonally planned city linked by paved roads, with aqueducts and sewages. Roman cities had a geometric order, a reticulate design.The Roman cities, with the exception of Rome, were divided into squares, based on the military camps called Castrum and in the centre was the Forum.
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As you can see, the city was surrounded by a wall (Murus) with watchtowers (Turris). The city is divided into modules, separated by parallel streets of equal dimensions. Two streets, however, have larger dimensions: the Cardus (direction N-S) and Decumanus (direction E-O), each leading to the four gates of the city. Where these two streets intersect is the Forum and the Mercatus, the most important spaces and buildings in the city. However, the city plan was generally rectangular, but could have the shape of another polygon.Loading...
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Bracara Augusta (Braga) a portuguese city in Roman timeConsidering the geometry of Roman cities, consider an Ideal City, where all streets are orientated from North to South or East to West. In addition, the streets are not wide and the buildings are the size of a point. In Ideal City there are no cars and people move either on foot or by bike. The distance between two buildings is measured in terms of the number of blocks that have to be traveled to get from one building to another. Thus, for example, in the figure, the distance between building P and building Q is 7.
If we define a Cartesian coordinate system, the distance between any two buildings
and
is given in Euclidean geometry by
However in the present case and taking into account the geometry of the Ideal City this distance is given by
This is a non-Euclidean geometry, the Taxicab geometry, revealed by Hermann Minkowski (1864-1909), German mathematician.
The geometric interpretation of this metric is given in the previous figure. While in Euclidean geometry the distance between points P and Q is given by the measurement of the dashed segment, in Taxicab geometry it is by the gathering of the measurements of the red segments.
The Taxicab geometry, one of the non-Euclidean geometries, is, to some extent, easy to see geometrically, as it can be employed in the urbanization of an ideal city, where the streets are organized in blocks, like Manhattan.
The geometric interpretation of this metric is given in the previous figure. While in Euclidean geometry the distance between points P and Q is given by the measurement of the dashed segment, in Taxicab geometry it is by the gathering of the measurements of the red segments.
The Taxicab geometry, one of the non-Euclidean geometries, is, to some extent, easy to see geometrically, as it can be employed in the urbanization of an ideal city, where the streets are organized in blocks, like Manhattan.
To better understand how this geometry works, I propose the following exercises:
1. On a sheet of squared paper representing the Ideal City, and choosing a Cartesian coordinate system, mark buildings A (- 2, - 1) and B (3, 2):
(a) determine d (A, B).
(b) find some buildings that are at distance 3 of A.
(c) represents all P buildings that are at distance of A.
(d) Represents all P buildings that are at distance of B.
(e) Represents the set of all buildings P which are at a distance 3 from both A and B.
2. Anne and Bob are looking for an apartment to live in the Ideal City. Anne works at City Museum M (- 3, - 1) and Bob at Post Office C (3, 3).
(a) Where should be located the apartment so that the distances traveled by Anne and Bob to move from home to their workplaces are exactly the same?
(b) Alternatively, Anne and Bob decide to look for an apartment in such a place that the sum of the distances they have to travel to get to their jobs is as small as possible. Where should the apartment be located?
(c) As a second alternative, Anne and Bob have decided to look for an apartment in such a way that the sum of the distances they have to travel to their jobs is less than or equal to 14. Where apartment should it be situated?
1. On a sheet of squared paper representing the Ideal City, and choosing a Cartesian coordinate system, mark buildings A (- 2, - 1) and B (3, 2):
(a) determine d (A, B).
(b) find some buildings that are at distance 3 of A.
(c) represents all P buildings that are at distance of A.
(d) Represents all P buildings that are at distance of B.
(e) Represents the set of all buildings P which are at a distance 3 from both A and B.
2. Anne and Bob are looking for an apartment to live in the Ideal City. Anne works at City Museum M (- 3, - 1) and Bob at Post Office C (3, 3).
(a) Where should be located the apartment so that the distances traveled by Anne and Bob to move from home to their workplaces are exactly the same?
(b) Alternatively, Anne and Bob decide to look for an apartment in such a place that the sum of the distances they have to travel to get to their jobs is as small as possible. Where should the apartment be located?
(c) As a second alternative, Anne and Bob have decided to look for an apartment in such a way that the sum of the distances they have to travel to their jobs is less than or equal to 14. Where apartment should it be situated?
3. The Mayor of the Ideal City is considering setting up a subway network to facilitate the movimentations in the city. It has already decided that the first line would connect (straight) the city center, where is House C (0, 0), to the Wharf of the Minkowski river R(- 6, - 2), which is the main means of communication of the city with the outside. Naturally, the president wants one of these stations to be built as close as possible to his house P (-3, 2). Where should be located such station be?
A final challenge...
Imagine that Peter, a Roman soldier, has three girlfriends, Anne, Bethy and Cammile. Of course, Peter wants to live as close as possible to the three. Representing A (- 3, 3), B (2, - 2) and C (8, 5), respectively, the buildings where the three girlfriends lives, what is the ideal place for Peter to live?
Imagine that Peter, a Roman soldier, has three girlfriends, Anne, Bethy and Cammile. Of course, Peter wants to live as close as possible to the three. Representing A (- 3, 3), B (2, - 2) and C (8, 5), respectively, the buildings where the three girlfriends lives, what is the ideal place for Peter to live?