# Revisualizing the Map of the Lower 48

Topology is a branch of mathematics that studies the properties of geometric objects that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending. In mapmaking, topology has to do with how features like points, lines, and polygons share geometry.

Now, what do you suppose might happen if those two points of view of what topology is were combined?

You don't have to wonder, because Topology Fact has taken a map of the lower 48 contiguous states of the U.S. and revisualized it. All state borders have been either stretched, twisted, crumpled, or otherwise bent as needed into rectangles, then positioned to indicate if they border each other or an ocean. Here's the result:

The topologist's map makes it easy to see some neat features that would be harder to see on a more traditional map. Such as:

• Maine is the only state that only borders just one other state.
• Four states, Maine, Rhode Island, South Carolina, and Washington only border two other states.
• Two states, Missouri and Tennessee, border eight other states each (and technically, each other).
• No state is more than four states away from an ocean.

The downside to all the stretching is that the topologist's map doesn't do well in depicting how large the states actually are with respect to each other. Texas has the largest area of all the states shown, but is tiny on the map. States like California (second-largest) and Arizona (fifth-largest) look almost identical in size, while Oregon is stretched to enormous proportion.

The map also suggests that both Arizona and New Mexico border an ocean. Both however are landlocked. That could be easily fixed however, if the rectangles for California, Arizona, New Mexico, and Texas were extended to the bottom edge of the map and the rectangle for Oregon shortened to indicate California has a beach.

The map also leaves open the question of how Alaska and Hawaii might be added to this kind of map. As created, they would be isolated islands separated from the rest. But then, that's exactly how wall maps of the United States in schoolrooms often display them.

If you want to take the map of the lower 48 states to another level of abstraction, mathmaticians can also turn to graph theory to revisualize how the boundaries of states are connected to each other. If nothing else, it makes it easier to count the number of borders for each state, but fully disconnects each from their geographic area.

# How Much Fossil Fuel CO2 Is in the Air?

Since the dawn of the industrial revolution in 1750 through 2021, people have burned enough coal, oil, natural gas, and other fossil fuels to add about 1,737 billion metric tons of carbon dioxide in the Earth's atmosphere.

But because of the Earth's carbon cycle, a large percentage of those emissions have been extracted from the air. Through 2021, land and sea-based carbon sinks have removed about 37% of total historic CO₂ emissions.

As a general rule of thumb, the older the fossil fuel emission, the more of it has been removed from the atmosphere, reducing the concentration of carbon dioxide in the air below what it would otherwise be without those natural processes. The following chart shows the history of cumulative fossil fuel emissions from 1750 through 2021, including the estimated amount of carbon dioxide cumulatively absorbed over that time.

Through 2021, about 642 billion metric tons of carbon dioxide have been removed from the Earth's atmosphere by the planet's natural carbon cycle. That leaves about 1,095 billion metric tons that have not.

The chart also shows a widening gap between the amount of carbon dioxide emitted from fossil fuels and the amount recovered through the carbon cycle. That's attributable to two factors. First, CO₂ emissions have been increasing rapidly, primarily due to China's growing carbon footprint over the last 30 years. Second, the newest emissions have had the least amount of time in which the natural carbon cycle can do its work.

That time matters. If you look closely at the chart to examine the cumulative emissions produced through 1990, you'll find that over half those CO₂ emissions are no longer present in the atmosphere. Because of the effect of the carbon cycle in absorbing older carbon dioxide emissions, the amount of carbon dioxide currently in the atmosphere represents more a consequence of what has been happening in the last 30 years than what happened in the more distant past.

The natural carbon cycle is only able to remove above 80% of these carbon dioxide emissions over a 300 year period. The remaining 20% takes much longer to absorb because it requires geologic processes that occur over thousands of years. Much of the carbon capture and storage (CCS) technologies in development today are being developed with an eye to reducing the initial amount of CO₂ emissions produced and to, in effect, mop up the long-term remainder.

### Analyst's Notes

Bonus chart! Because the featured chart illustrates the exponential growth of fossil fuel-based carbon dioxide emissions that spans several orders of magnitude, here's a link to a second chart presenting the same data using a logarithmic scale. We opted to feature only the chart showing the data on a linear scale since it better illustrates the relative percentage share of absorbed carbon dioxide emissions with respect to the total.

### References

Friedlingstein et al. Global Carbon Budget 2022, Earth System Science Data, 11 November 2022. DOI: 10.5194/essd-14-4811-2022.

Political Calculations. How Long Does Carbon Dioxide Stay in the Atmosphere? [Online Article, Tool]. 19 July 2023.

Image credit: National Oceanic and Atmospheric Administration. Carbon Cycle Illustration. 2019. Public Domain Image.

# The S&P 500 Visualized as a Treemap

What's the best way to visualize the relative size of 505 stocks with respect to each other?

By size, we're referring to the market capitalization of a company's stock, which is the product of its price per share and the number of shares it has issued. And by 505 stocks, we're referring to the stocks of the companies that make up the S&P 500 (Index: SPX).

Since the contribution of each of its component stocks are weighted according to their market cap, being able to visualize how its component firms compare to each other provides valuable information for investors. If you're paying attention to the value of the index, you can use the information to quickly determine which individual stocks can most affect the index as their stock prices change.

Previously, we were impressed by how Visual Capitalist visualized the S&P 500, but now, we've come across another visualization that follows a similar approach, but delivers more interactive and timely information for investors/information consumers. Finviz, a financial information site that visualizes the relative size of the component firms that make up the S&P 500 index using a treemap chart.

Like the bubble chart-style presentation Visual Capitalist used, Finviz' treemap of the S&P 500 groups firms from the same industrial sectors together. What takes Finviz' presentation to the next level however is their chart is updated daily, indicating how much each component company's market cap changed from the previous trading day. We took a snapshot of Finviz' S&P 500 treemap after the end of trading on Friday, 4 August 2023, the day after the 2023-Q2 earnings announcements for Apple (NASDAQ: AAPL and Amazon (NASDAQ: AMZN. As you can quickly see from the snapshot, it was a bad day for Apple and the S&P 500 index overall, but a very good day for the smaller market cap Amazon.

Finviz' treemap presentation is also interactive. You can zoom in on the S&P 500 treemap chart at their site to focus on smaller firms within a particular industrial sector. Meanwhile, clicking on a sector of the treemap will open a popup window presenting sparkline charts of each sector component firms' stock prices over the past three months, so you can quickly get both its current share price and see how each share price has been trending.

All-in-all, we think Finviz' approach for visualizing and presenting this data is very well done.

# Visualizing the Entire S&P 500

One of the biggest challenges market analysts have is how to visualize the companies that make up the S&P 500 (Index: SPX). These firms range in size from the largest, Apple (NASDAQ: AAPL), which at this writing has grown to a market capitalization of around \$3 trillion, to the smallest, Fortrea Holdings (NYSE: FTRE), which just barely qualified to make it into the index with a market cap of \$2.85 billion after being spun off from Labcorp (NYSE: LH) last week.

Altogether, there are 505 firms that make up the S&P 500. There have been since January 2015 when Standard and Poor included additional share classes of five S&P 500 firms whose own market capitalizations met the index' requirements for inclusion.

That very big difference between the market capitalization of the biggest and smallest firms of the market-cap weighted index makes it very difficult to visually compare the firms that make up the S&P 500. Rising to the challenge, Visual Capitalist's Dorothy Neufield has worked out a useful approach, grouping the firms within their industrial sector to illustrate their relative sizes. Here's her chart, which represents a snapshot of the index as it existed on 5 May 2023:

She's since followed up with another visualization featuring the most profitable companies within each industrial sector. Before you click over, see if you can guess which firms those are from their relative sizes in this visualization.

### Reference

Dorothy Neufield. Visualizing Every Company on the S&P 500 Index. Visual Capitalist. [Online Article]. 21 June 2023.

# Visualizing 52 Years of U.S. Mortgage Rates

In the United States, 30-year mortgage rates play a big role in determining how affordable housing is for American households.

As important as they are however, the historical data for mortgage rates is surprisingly lacking. Freddie Mac, officially known as Federal Home Loan Mortgage Corporation, only has weekly data extending back to April 1971. The government-sponsored enterprise used to report monthly averages for mortgage rates, but that discontinued that practice after December 2022.

And yet, because housing sales and prices are reported on a monthly basis, it's incredibly useful to have mortgage rates averaged over the period of a month. Since Freddie Mac isn't doing that job any more, we're taking it over. Not only that, we're making all that historic data freely available!

It's built into the following interactive chart, which visualizes 52 years worth of the average monthly interest rates for 30-year conventional mortgages in the U.S.

Some quick observations:

• In April 1971, the average 30-year mortgage rate was 7.31%.
• The hyperinflation of the late 1970s contributed to the rapid rise of mortgage rates, which first peaked at 16.33% in April 1980, then fell back to 12.19% just three months later. Mortgage rates then proceeded to rise rapidly once more, reaching a record high peak of 18.45% in October 1981.
• Over the next 40 years, the 30-year conventional mortgage rate generally fell in a long, sustained downtrend. The interest rate on a 30-year conventional mortgage ultimately bottomed at 2.77% in November 2020.
• Mortgage rates then began rising slowly, reaching 3.10% in December 2021.
• After December 2021, mortgage rates surged as the Biden-era's inflation firmly took hold. The average mortgage rate peaked at 6.90% in October 2022, before falling back. During 2023, the interest rate on a conventional 30-year mortgage has ranged between 6.27% and 6.54%. In April 2023, the average 30-year mortgage rate was 6.34%.

We plan to update the chart every few months. If you are looking to find a recent month's average rate that hasn't yet been included, see the references....

### References

Freddie Mac. 30-Year Fixed Rate Mortgages Since 1971. [Online Database]. Accessed 23 May 2023. Note: Starting from December 2022, the estimated monthly mortgage rate is taken as the average of weekly 30-year conventional mortgage rates recorded during the month.

Image credit: Mortgage Payment Due date by alanharder.ca via Wikimedia Commons. Creative Commons Attribution 2.0 Generic (CC BY 2.0).