There are and have been different approaches to this question in my view. If you consider the historical perspective, there will certainly appear the orthogonal layouts e.g. of planned roman, mayan, chinese and  khmer cities. Later on you find the planned "ideal cities" with different forms due to the purpose of the planning (representation, renaissance-thinking, innovation-driven).

The geometric layout of a city is mostly a product of pre-existing routes, joining in one point and thus leading to a certain geometry. The ring-radial system can be seen as the most useful (most often seen) geometry in terms of connectivity, orientation, centrality and growth-ability.

The geometry depends on the topography, laws and technology available and is a product of various external influences in time. By simplifying the urban geometry to point, line and area, we can get the different forms and combinations of forms found all over the world.

The optimal geometry is the one which suits best to the topography, geomorphology, hydrology and local climate and can provide a structure for the cultural, aesthetical and sustainable perception of the people who live there.

Further perspectives could be: Space Syntax (see 1st link), Urban Morphology (2nd link), Settlement Structure, Spatial Network-Theory

http://www.sss7.org/Proceedings/01%20Key-note%20Papers/K01_Hillier_Spatial_Sustainability.pdf

http://www.urbanform.org/

Dear Benjamin,

Thanks for your insightful answers and the research leads you have included.

As Benjamin Casper very relevant indications focus on some detailed description of the form and of the networks, the discussion is going a bit away from your initial question more focussing on the external envelop of the urban form.

I would simply add to the discussion at this stage the idea of fractals that are a way to describe some rather complex forms as large cities are. As I understand your interest is on geometry, this is worth considering, and also fractals come with measurements and comparison between forms, so they can indicate some elements for discussing optimality.

In France we have a specialist of these issues, Pierre Frankhauser, I attach one of his numerous papers.

Article Fractal Dimensions of the Built-up Footprint: Buildings vers...

Dear Shian-Loong,

The following suggestions have based on the problems of established cities:

http://www.fractalcities.org/book/Fractal%20Cities%20Chapter%201.pdf

http://eandt.theiet.org/magazine/2009/21/model-for-cities.cfm

http://www.bbc.co.uk/news/technology-21032725

http://www.reddit.com/r/TrueAskReddit/comments/2iegyf/how_differently_would_we_design_cities_from/

 I hope to collaborate.

Best Regards,

Vanessa

Dear Alain,

Your views has given me an altogether different view of fractals.As before I used to zoom in on the self-similarity aspects-being unaware of the metrics between forms.Also my previous tendency was to interpret fractal optimality as "nature optimality" - how amendable fractal forms are to the geometry of nature. Thanks for the link as well.

Dear Vanessa,

Indeed the problems with urban form are governance-related and institutional as well. It would be interesting to know how planners have incorporated historical understanding and more formal approaches in their assessments of urbanism.

   As Alain L'Hostis wrote, the fractal approach is insightful regarding geometrical issues. However, it may complicate your discussion of optimality. I would just add the following reference, still involving I. Thomas and P. Frankhauser, that proposes a microeconomic urban model within a fractal space, thus allowing to discuss optimality issues, especially regarding green places accessibility.

On publisher's website:

http://www.envplan.com/abstract.cgi?id=a36126

On ResearchGate:

https://www.researchgate.net/publication/23539461_Where_Alonso_meets_Sierpinski_an_urban_economic_model_of_a_fractal_metropolitan_area

Article Where Alonso Meets Sierpinski: An Urban Economic Model of a ...

Geometry for my view evolves from series of triangular networks, which takes the shape to an octagonal form in terms of relationships between the nodes. To construct the form of cities one needs to connect the nodes (i.e dots) which evolves from historical settlement patterns. The in-between spaces are connects established through the land uses. The optimization emerges from the purpose of form construction of geometry you need to establish between the hierarchy of nodes and connections (roads). Every town has emerged as different fractal formats given the activities concentrated on the nodes. There is connectivity established through the state of urban economy, mode of transport and technology usage. Class of cities and geographical limits tends to pre-define the geometry however, there is deliberate attempt by the planners to follow certain pattern for the ease of services-public utilities. Its interesting to recreate the cities from different continents, regions, and economic categories I am sure you would find interesting geometric formulations. Good Luck 

DearJustin and Shashikant,

Thanks for the info and views- much appreciate the diversity of perspective on the question

Read vol 1 by Les King, and perhaps the other monographs in the Scientific Geography Series. Now available as a no cost PDF at "Classics In Regional Science"

http://rri.wvu.edu/resources/web-book-rs/classics-in-regional-science/

http://rri.wvu.edu/resources/web-book-rs/classics-in-regional-science/

Read my book Land Use and Urban Form on my general method which explains the general spatial market processes. The book can be obtained at no charge as a PDF at the "Classics In Regional Science" web site 

http://rri.wvu.edu/wp-content/uploads/2012/12/Land-Use-and-Urban-Form.pdf

http://rri.wvu.edu/wp-content/uploads/2012/12/Land-Use-and-Urban-Form.pdf

In my opinion there is not an optimal geometry you can apply to the kind of urban complex. Each agglomeration has its own particularities that will ruin a geometrical plan. I would take in account the structuralist urban planners than the modernist ones.   

Dear Juan,

Yes, I also suspect that a geometric approach may not be an end in itself but as more of an indicator of a more pervasive underlying process that is occurring- hopefully that can be captured in an index.

Best Regards,

Bernard

Dear  Gerhard,

Thanks for outlining the theoretical bounds on "optimal geometry" and for your very interesting lead to your work on nets. Perhaps the "universum" of nets can serve as a template that has in it a continuum of optima for the topology we seek to fit to the urban structures encountered in reality?

Dear Bimo,

Your picture brings to mind nested structures- which is in a way a net within a net?

Reminding me of the great a sociobiologist's lifework (and the current fad with biomimicry) with ants. How often we pit nature against nurture- when sometimes we need to embrace both

Best Regards,

Bernard

http://www.hup.harvard.edu/catalog.php?isbn=9780674002357

@ Alain... The artist view on fractals in urban maps...

I thought I share this different perspective on urban "geometry" here.

http://www.citylab.com/design/2015/03/how-fractals-bring-imaginary-cities-to-life/386815/

Hi Benjamin,

A picture is worth a thousand words...

Tight budgets need not be a limitation to good desgins

http://www.vmwp.com/