Halfday Trip to Western Cleveland

This trip is best suited for those in the southeastern suburbs of Cleveland, and takes maximum advantage of Cleveland RTA's 2.5-hour free transfer, to quick exploration of Edgewater Park and Westside Market, both on city's west side. To start out, pick a day with weather like today's: sunny and low 70s. For this trip, bring these items and drive to the Van Aken Blvd (or Green Road) rail terminus.

• 5-ride RTA pass
• book to read
• scooter (optional)

While the 5-ride pass is $11.25, it's ideal if there's plan to use it in the future before the expiration date. The one I bought today doesn't expire until start of 2014. A book to read is ideal for the train ride there and back. Unlike the New York City Subway, there will very likely be available seats, particularly in non-rush hours. Scooter may assist in getting around after getting off the train.

From Van Aken Blvd station, ride the entire length of the rail-line to Tower City. It isn't until here that one needs to swipe the RTA pass and the 2.5 hour period begins. Hop over to the Red Line and go westbound for 3 station to West Blvd, in the neighborhood area of Cudell/Edgewater, a community that "[has] the appeal of a suburban community, but offer convenient proximity to downtown, and have churches and entertainment within walking distance." Walk/scooter for about 10 minutes down to the beach northward via West Blvd. Surely enough, the stretch on West Blvd is a quiet area with apartments and single-family houses.

A day like this is the best to soak up the sun at the beach. Afterwards, head back to the rail station. Ride 2 stations eastbound to W. 25th Street and arrive at the historic West Side Market, whose origins date back to 1840 and present attract nearly a million visitors annually. On this date, watermelon was available, although in scant quantities and selling for as high as $12. Nevertheless after a quick meal there, a quick ride back to Tower City and transferring onto Blue (or Green) line takes back to the start.

In all, the trip consisted of 5 different rides and 2 different destinations. But thanks to the unlimited transfer within the 2.5 hour period, all of the rides fell within the one $2.25 fare.

Sources:

• http://www.livecleveland.org/cudellEdgewater
• http://www.westsidemarket.org/about.html

Yonkers Loop Biking Adventure

From Columbia University, the Hudson Greenway goes up to Dyckman Street. The subsequent paths in Inwood Hill Park may get confusing, but there supposedly is a bike path over Henry Hudson Bridge that cyclists can walk their bikes across. However, unable to find the entrance to that path to cross Spuyten Duyvil Creek, I crossed Harlem River Ship Canal via Broadway Bridge instead. This subsequent stretch on Broadway doesn't contain bike paths, but convenient has a wide lane for parked cars, and cyclists can squeeze in with enough space. Continuing northbound, Broadway passes all of the remaining stations of the 1 train and the western edge of Van Cortlandt Park. After W 262nd Street, Broadway enters Yonkers as South Broadway.

The transition is seemingly noticeable. Continuing S. Broadway in Yonkers, the area is highly reminiscent of area in The Bronx. Upon encountering Main Street, downtown Yonkers is approached. The Yonkers Metro North Railroad station is located here, as well as the Waterfront. This area is directly across Hudson River from Alpine, NJ and about 7 miles north of George Washington Bridge. Returning southbound, Riverdale Ave doesn't feature many uphills. The route enters The Bronx and ends at W 230th Street. From that point, Broadway - Dyckman - Hudson Greenway takes back to Morningside Heights for an overall of nearly 24 mile trip, devoid of major difficult uphills.

Sources:

• http://www.mapmyride.com/routes/view/73761434
• http://www.dnainfo.com/20100811/washington-heights-inwood/cyclists-cant-ride-across-henry-hudson-bridge-despite-millions-spent-on-bike-path-renovation

High Population-Density Areas

A recent New York Times articled entitled "How Many People Can Manhattan Hold?" discussed the population and capacity of Manhattan. While only 1.6 million people call Manhattan home, population in the city more than doubles and may reach as high as 3.9 million during the weekday, and during special events like Thanksgiving Day Parade, may reach as high as 5 million. While Manhattan and New York City may easily be the densest place in the United States, how does it compare to other metropolises of the world?

Initial look at ranking charts reveals that predominantly Asian cities top the rankings for the most dense cities. However, comparison with other metropolises eclipses the fact that Manhattan's figure is much greater than that of New York City, and worldly comparisons usually take into account of the entire city proper. With over 8 million residents and spanning over 300 square miles, New York City has about 27,000 people / square mile. However when Manhattan is solely focused, the 1.6 million people in land area of around 23 square miles corresponds to about 70,000 people / square mile. That figure eclipses the population densities of metropolises like Delhi, Mumbai, Calcutta. However, caution should be noted that while Manhattan is simply a dense enclave of New York City with population around 1.6 million, the figures for the Indian metropolises include the entire city proper that has 5 to 10 million people.

A discussion of high population density cannot go unfinished without mentioning Kowloon Walled City. A former Chinese garrison area and a residential area in Hong Kong before demolished in 1993, Kowloon Walled City had about 33,000 people living in the 6.5 acre area, corresponding to over 3 million people / square mile. At that density, 65 million people could be jammed into Manhattan. Even if "the current streets and parks were left intact," about half of that, or the population of California could fit into Manhattan.

Sources:

• http://www.nytimes.com/2012/03/04/realestate/how-many-people-can-manhattan-hold.html
• http://www.forbes.com/2007/12/14/cities-pollution-asia-biz-logistics-cx_tvr_1214densecities.html
• http://quickfacts.census.gov/qfd/states/36/36061.html
• http://chom.hubpages.com/hub/The-densest-living-place-in-the-world-Hong-Kong-Kowloon-Walled-City
• http://www.tofu-magazine.net/newVersion/pages/KWC.html

Approximation of Natural Log

Taking the Taylor series of ln(1+x) about x=0 gives the following result, as demonstrated by the MATLAB line:

>> syms x;
>> taylor(log(1+x))
ans =
x^5/5 - x^4/4 + x^3/3 - x^2/2 + x

The common approximation is that as long as x is small, ln(1+x) ~= x. How close is this approximation anyways? In the following Excel table, various small and incrementing values of x were chosen, along with the actual ln(1+x) values. In all cases, the approximation yields a slightly higher answer, since the most significant term of -x^2 was left out. The difference and the % error to the actual values were calculated:

x	ln(1+x)	+ / -	% Error
0.000	0.00000	0.00000	#DIV/0!
0.005	0.00499	0.00001	0.250%
0.010	0.00995	0.00005	0.499%
0.015	0.01489	0.00011	0.748%
0.020	0.01980	0.00020	0.997%
0.025	0.02469	0.00031	1.245%
0.030	0.02956	0.00044	1.493%
0.035	0.03440	0.00060	1.740%
0.040	0.03922	0.00078	1.987%
0.045	0.04402	0.00098	2.233%
0.050	0.04879	0.00121	2.480%
0.075	0.07232	0.00268	3.705%
0.100	0.09531	0.00469	4.921%
0.200	0.18232	0.01768	9.696%
0.300	0.26236	0.03764	14.345%
0.400	0.33647	0.06353	18.881%
0.500	0.40547	0.09453	23.315%
0.750	0.55962	0.19038	34.021%
1.000	0.69315	0.30685	44.270%

Observing for x is 0.05 or smaller, it seems as though the absolute error grows in a quadratic manner, while the percentage error grows constantly. But most importantly, if the approximation of ln(1+x) = x wants to stay within 5% of the actual answer, the value of x shouldn't exceed 0.1.

Maps

Earlier this week, two maps that I had ordered from the New York State Department of Transportation arrived in my mailbox: Long Island Biking Map and New York State Biking Route 9 Map, joining my walls that already felt inundated with 6 other maps. Now, I have barely been able to complete a Coney Island round-trip in an afternoon. With so many parts of the actual New York City (mainly Queens) yet to be traveled through, I really have no immediate plans of any excursions towards Montauk or Rouses Point. But maps aren't just meant for going on adventures. Maps are direct ways to understand and connect with the surroundings.

When I am asked why I like biking, I usually give the straightforward reasons of exercise, monetary savings, and the ability to claim credits for an adventure. Adventure is simply a way to experience the understanding of the surrounding  place. While motorized vehicles certain facilitate traveling, doing so without the external assistance makes the journey feel all the more relatable. That is, it allows for a better connection with the routes and the places. After all, how can I not associate Music Street just east of OH-306 as the place where I was stranded after my bike tire popped? And by planning and going on the actual routes, it sure helps to understand the area.

And maps help fill in where direct adventures aren't possible. Even if there isn't the actual travel, there can be the passive observation and understanding: the knowledge that NY Biking Route 27 leads to the Montauk Lighthouse at the eastern tip of Long Island, or that Route 9 travels along the Hudson Valley all the way up to the Canadian border. One day, it may be directly useful. But even if not, it's just a way to appreciate the capabilities and possibilities out there.

Understanding geography and locations are much simpler than understanding the current events. In the words of the 10-year-old myself while traveling down to Florida from Ohio, "I just wanted to get out and take a step [on Virginia] to know that I was indeed there!" Direct experience makes it so much more realistic. In the absence of it, maps bridge the gap to bring the understanding of the surrounding in a more direct way. Understanding the outside surroundings is a key to engage in and be part of  where one is. That's why I usually leave AM 1130 Bloomberg Radio (or WTAM 1100 back in Cleveland) on. While the radio dynamically feeds in information, the 8 maps saturating the wall of my dormitory room help to illustrate where everything stands. That order in itself is a source in its own.