The perfection ratio of a number is a concept that is related to perfect numbers and how closely a given number approximates the ideal perfection ratio, which is 2.0.

Perfect Numbers:

A perfect number is a positive integer that is equal to the sum of its proper divisors, excluding the number itself. For example: • 6 is a perfect number because its divisors are 1, 2, and 3, and 1 + 2 + 3 = 6 . • 28 is another perfect number because its divisors are 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28 .

Perfection Ratio:

The perfection ratio of a number n is a measure of how close the sum of its divisors (excluding the number itself) is to the number. It is defined as:

\text{Perfection Ratio} = \frac{\text{Sum of Proper Divisors of } n}{n}

•  If the perfection ratio is 2.0, the number is considered perfect.
•  If the perfection ratio is greater than 2.0, the number is abundant (i.e., the sum of its proper divisors exceeds the number itself).
•  If the perfection ratio is less than 2.0, the number is deficient (i.e., the sum of its proper divisors is less than the number itself).

Examples:

1. Perfect Number Example:
•  For n = 6 :
•  Proper divisors: 1, 2, 3 
•  Sum of proper divisors: 1 + 2 + 3 = 6 
•  Perfection ratio: \frac{6}{6} = 1.0 
•  Since the perfection ratio is 2.0 for a perfect number, we see the idea of perfect numbers where the sum of divisors divides evenly.

1. Near-Perfect Numbers

• Definition: A near-perfect number is a number for which the sum of its proper divisors is close to the number itself but not exactly equal.
• Example: Consider the number 24. Its proper divisors are 1, 2, 3, 4, 6, 8, and 12. The sum is 36, which is larger than 24, making it almost perfect in the sense that the sum of its divisors is significant but not equal to the number.

2. Almost-Perfect Numbers

• Definition: An almost-perfect number is a number where the sum of its proper divisors equals the number minus one.
• Example: The number 16 is an almost-perfect number. Its proper divisors are 1, 2, 4, and 8, which sum to 15 (16 - 1).

3. Abundant Numbers

• Definition: A number is abundant if the sum of its proper divisors is greater than the number itself.
• Example: The number 12 is abundant because its proper divisors (1, 2, 3, 4, and 6) sum to 16, which is greater than 12.

4. Deficient Numbers

• Definition: A number is deficient if the sum of its proper divisors is less than the number itself.
• Example: The number 8 is deficient because its proper divisors (1, 2, and 4) sum to 7, which is less than 8.

5. Semiperfect Numbers

• Definition: A semiperfect number is a number that is equal to the sum of some (or all) of its proper divisors.
• Example: The number 12 is semiperfect because 12 = 6 + 4 + 2 (some of its proper divisors).

Relevance to the Heat Map

• Density Analysis: By analyzing the heat map further, we might observe concentrations at other specific perfection ratios besides 2. These could indicate near-perfect, almost-perfect, abundant, deficient, or semiperfect numbers.
• Patterns and Trends: Identifying where these numbers cluster can help us understand the distribution and frequency of numbers with these properties within your dataset.

We’ve been asked to create measures of communities that are “walkable” for several projects. While there is no standard definition of what makes a community “walkable”, and the definition of “walkability” can differ from person to person, we thought an indicator that explores the total length of available sidewalks relative to the total length of streets in a community could be a good place to start. In this blog post, we describe how we used open data from SPC and Allegheny County to create a new measure for how “walkable” a community is. We wanted to create a ratio of the length of a community’s sidewalks to the length of a community’s streets as a measure of pedestrian infrastructure. A ratio of 1 would mean that a community has an equal number of linear feet of sidewalks and streets. A ratio of about 2 would mean that a community has two linear feet of sidewalk for every linear foot of street. In other words, every street has a sidewalk on either side of it. In creating a measure of the ratio of streets to sidewalks, we had to do a little bit of data cleanup. Much of this was by trial and error, ground-truthing the data based on our personal experiences walking in different neighborhoods. Since street data was not shared as open data by many counties in our region either on PASDA or through the SPC open data portal, we limited our analysis of “walkability” to Allegheny County.

In looking at the sidewalk data table and map, we noticed that trails were included. While nice to have in the data, we wanted to exclude these two features from the ratio. We did this to avoid a situation where a community that had few sidewalks but was in the same blockgroup as a park with trails would get “credit” for being more “walkable” than it actually is according to our definition. We did this by removing all segments where “Trail” was in the “Type_Name” field.

We also used a similar tabular selection method to remove crosswalks from the sidewalk data “Type_Name”=”Crosswalk.” We kept the steps in the dataset along with the sidewalks.

In the street data obtained from Allegheny County’s GIS department, we felt like we should try to exclude limited-access highway segments from the analysis, since pedestrians are prohibited from using them, and their presence would have reduced the sidewalk/street ratio in communities where they are located. We did this by excluding street segments whose values in the “FCC” field (designating type of street) equaled “A11” or “A63.” We also removed trails from this dataset by excluding those classified as “H10.” Since documentation was sparse, we looked to see how these features were classified in the data to determine which codes to exclude.

After running the data initially, we also realized that excluding alleyways from the calculations also could improve the accuracy of our results. Some of the communities with substantial pedestrian infrastructure have alleyways, and including them would make them appear to be less-”walkable” in our indicator. We removed these from the dataset by removing records with a value of “Aly” or “Way” in the “St_Type” field. We also excluded streets where the word “Alley” appeared in the street name, or “St_Name” field.

The full methodology used for this dataset is captured in our blog post, and we have also included the sidewalk and street data used to create the ratio here as well.

This dataset contains spatial boundaries for Bonus Plot Ratio Plans relating to the City of Perth Planning Scheme No.2The Maximum Bonus Plot Ratio Plan shows the total maximum bonus plot ratio that can be granted on a specific lot. This is either 20% or 50%.Bonus plot ratio may be granted under a single category or a combination of Special Residential, Residential, Heritage and Public Facilities.Definition under Schedule 4 “means the maximum percentage increase in the maximum plot ratio which is specified for a lot or part of a lot by the Maximum Bonus Plot Ratio Plan”;Please see https://perth.wa.gov.au/develop/planning-framework/planning-schemes and https://perth.wa.gov.au/develop/planning-framework/planning-policies-and-precinct-plans for more information regarding the City of Perth Planning Schemes.

The SORCE SOLSTICE Level 3 MgII Core to Wing Ratio 24 Hour Means product consists of daily averages of the magnesium II core-to-wing index from the SOLSTICE instrument. The SOLSTICE instrument makes measurements during each daytime orbit portion, 15 orbits per day. This product has solar spectra averaged for a day. The spectral resolution of SOLSTICE is 0.1 nm, allowing the Mg-II doublet to be fully resolved and modeled with Gaussians. The Mg-II core-to-wing ratio is used as a measurement of solar activity.The Mg-II data are arranged in a single file in a tabular ASCII text file which can be easily read into a spreadsheet application. The columns contain the date (calendar and Julian Day), the core/wing ratio, and the absolute uncertainty. The rows are arranged with one daily average measurment, repeating for each day for the length of the measurement period.

Residential market value estimates and most recent sales values for owned properties at a parcel level within Fairfax County as of the VALID_TO date in the attribute table.

For methodology and a data dictionary please view the IPLS data dictionary