Here's a challenge. Find a three dimensional sphere whose surface iscovered by intricate patterns. Now, figure out how to project those patterns onto a two-dimensional sheet of paper, so that the resulting image remains as true as possible to the original patterns as they appear on the sphere.

Solving that exact geometrical problem has challenged mapmakers for centuries, because no matter how you project a 3D surface onto a 2D one, you'll end up distorting the true picture. In the following six minute video, Vox explains why all world maps get it wrong:

Mathematicians have an answer for minimizing that problem, which involves treating the surface of the globe you want to project onto a 2D surface like an orange peel. In the following Numberphile video, Hannah Fry demonstrates that concept, called the Euler Spiral, which takes 13 minutes to explain, and which requires scissors and a lot of patience to execute.

While the Euler Spiral represents what might be the optimal way to convert a three dimensional spherical surface into a two dimensional representation with minimal distortion, there is a new method for presenting a world map that trades its complexity for relative simplicity and a surprisingly small increase in distortion.

For centuries, mapmakers have agonized over how to accurately display our round planet on anything other than a globe.

Now, a fundamental re-imagining of how maps can work has resulted in the most accurate flat map ever made, from a trio of map experts: J. Richard Gott, an emeritus professor of astrophysics at Princeton and creator of a logarithmic map of the universe once described as "arguably the most mind-bending map to date"; Robert Vanderbei, a professor of operations research and financial engineering who created the "Purple America" map of election results; and David Goldberg, a professor of physics at Drexel University.

Their new map is two-sided and round, like a phonograph record or vinyl LP. Like many radical developments, it seems obvious in hindsight. Why not have a two-sided map that shows both sides of the globe? It breaks away from the limits of two dimensions without losing any of the logistical convenience—storage and manufacture—of a flat map.

Better still, here's a very short video that presents their mapmaking innovation:

The way they did it was to first score existing maps for six different types of distortions, with a score of 0.0 representing a perfect representation. They then applied their expertise in polyhedra geometry, the shapes of multi-sided solids, to work out how to project the world's 3D surface onto them to beat the best score of their cartographic competition. But then, their work took on a very different shape:

In a recent paper, Gott began considering "envelope polyhedra," with regular shapes glued together back-to-back, which led to the breakthrough idea for the double-sided map.

It can be displayed with the Eastern and Western Hemispheres on the two sides, or in Gott's preferred orientation, the Northern and Southern Hemispheres, which conveniently allows the equator to run around the edge. Either way, this is a map with no boundary cuts. To measure distances from one side to the other, you can use string or measuring tape reaching from one side of the disk to the other, he suggested.

"If you're an ant, you can crawl from one side of this 'phonograph record' to the other," Gott said. "We have continuity over the equator. African and South America are draped over the edge, like a sheet over a clothesline, but they're continuous."

This double-sided map has smaller distance errors than any single-sided flat map—the previous record-holder being a 2007 map by Gott with Charles Mugnolo, a 2005 Princeton alumnus. In fact, this map is remarkable in having an upper boundary on distance errors: It is impossible for distances to be off by more than ± 22.2%. By comparison, in the Mercator and Winkel Tripel projections, as well as others, distance errors become enormous approaching the poles and essentially infinite from the left to the right margins (which are far apart on the map but directly adjacent on the globe). In addition, areas at the edge are only 1.57 times larger than at the center.

That's a lot of problems solved to produce a reasonably simple 2D representation of the world that both minimizes distortion and remains useful as a map. Even better, their innovative concept may be patentable:

To the best of their knowledge, no one has ever made double-sided maps for accuracy like this before. A 1993 compendium of nearly 200 map projections dating back 2,000 years did not include any, nor did they find any similar patents.

"Our map is actually more like the globe than other flat maps," Gott said. "To see all of the globe, you have to rotate it; to see all of our new map, you simply have to flip it over."

At this writing, a U.S. patent application for the invention of the world disk map has yet to be filed. The Inventions in Everything team will periodically check to see if that status changes.

References

David M. Goldberg et al. Flexion and Skewness in Map Projections of the Earth, Cartographica: The International Journal for Geographic Information and Geovisualization (2007). DOI: 10.3138/carto.42.4.297

Envelope Polyhedra: arxiv.org/abs/1908.05395

Median new home prices in the U.S. have decoupled from median household income. Since May 2020, new home prices have risen sharply while median household income has largely stagnated.

The following chart reveals that recent development in the relationship between the trailing twelve month averages of median new home sale prices and median household income.

That combination of factors means that the typical new home built in the U.S. has become relatively more unaffordable for the typical American household.

Picking up on recent history, after peaking at 5.45 times (5.45X) median household income in February 2018, the relative level of affordability of new homes in the U.S. improved steadily as median household incomes rose much more quickly than new home prices. The ratio bottomed at 4.88X median household income in May 2020.

However, a surge of demand after May 2020 has fueled rising prices for new homes, which have rapidly risen to 5.08X median household income in January 2021. The timing of the surge of demand suggests its cause can be found in the responses of state and local governments to the coronavirus pandemic and to the outbreaks of anti-police protests.

Many states and local governments have periodically imposed lockdowns in their bids to slow or reverse the spread of coronavirus infections, in which many businesses are forced to close or to severely limit their operations while residents are ordered to stay at home. Several high population states have imposed especially severe lockdown measures, such as California, Illinois, New York, New Jersey, and Pennsylvania, which have generated population migrations out from these states by businesses and people seeking greater economic freedom. That migration benefits states that have not imposed similarly restrictive lockdowns, such as Arizona, Texas, and Florida, whose economic climates have suffered far less self-inflicted damage from imposing such measures. The lockdown measures themselves appear to have done little to contain the spread of coronavirus infections during the pandemic.

At the local level, the breakdown of public order has affected several large cities following local officials' actions to defund police departments and to reduce criminal prosecutions in the aftermath of the Black Lives Matter anti-police protests and riots at the end of May 2020 and early June 2020. Cities such as Chicago, Minneapolis, Los Angeles, New York, San Francisco, Portland and Seattle, have each experienced significant increases in crimes against people and property as they have become more unsafe, as would be predicted. The resulting decrease of public order in these cities is driving outward migration flows from them, boosting the demand for new homes outside the newly crime-infested jurisdictions.

The consequences of these things together result in the decoupling of median household sale prices from median household income, as the increased demand for new homes is paired with falling incomes.

How long that state of affairs will continue remains to be seen. With most of the responsibility for these conditions lying with elected state and local government officials in high population states, it may well continue until they reopen their economies, lift their stay at home orders, reestablish criminal law enforcement as a priority, or are replaced in office by those who will.

February is typically a strong month for dividend paying stocks in the U.S. stock market. That's because it is a very popular month for firms to announce they will increase their dividends based on the strength of their fourth quarter earnings.

So it's no surprise that February 2021 was a relatively strong month! As Februaries go however, February 2021 saw the lowest number of dividend rises announced in any February of the past eight years. The latest update to our chart illustrating the number of dividend rises and reductions in each month since January 2004 shows how February 2021 ranks for both metrics.

Let's get to February 2021's dividend metadata. Before we proceed however, we need to note that Standard and Poor did indeed revise their tally of the number of extra (or special) dividend payouts announced during January 2021, which turned out to be a considerably larger number than the zero they originally reported! The revised figure is the number we'll be referencing for January 2021's extra dividends below:

• A total of 2,439 U.S. firms declared dividends in February 2021, an increase of 973 over the unchanged number of 1,466 recorded in January 2021. That figure is also a decrease of 1,243 from the 3,682 recorded in February 2020.
• 64 U.S. firms announced they would pay a special (or extra) dividend to their shareholders in February 2021, a decline of 49 from the 113 recorded in January 2021 and an increase of 17 over the 47 recorded in February 2020.
• 286 U.S. firms announced they would boost cash dividend payments to shareholders in February 2021, an increase of 112 over the number recorded in January 2021, and a decrease of 11 from the 297 recorded in February 2020.
• A total of 14 publicly traded companies cut their dividends in February 2021, a decline of 18 from the 32 recorded in January 2021 and also a decrease of 15 from the 29 recorded in February 2020.
• 1 U.S. firm omitted paying their dividends in February 2021, a decline of 2 from the 3 recorded in January 2021. That figure is also a decrease of 3 from the 4 recorded in February 2020.

Meanwhile, a record 21 firms announced they would resume paying dividends in February 2021 after having suspended (or omitted) paying them during the coronavirus recession of 2020. This is five more than the previous monthly record of 16, which was recorded twice before, in December 2008 and in November 2020.

Since it has only been two weeks since we last reviewed the dividend cutting firms we've found in our ongoing real-time sampling of dividend declarations, we'll simply note the total dividend cuts announced in the first quarter of 2021 is continuing to run well below the pace of the first quarters of the previous three years. We'll next update the list after the end of March 2021, when we present the next update in the Dividends By The Numbers series!

In the meantime, should S&P revise its posted dividend data for February 2021, we'll append the version of this article that appears on our site with that information. Just like we had to for January 2021.

Political Calculations' initial estimate of median household income of in January 2021 is $66,039, an decrease of $241 (or 0.4%) below the initial estimate of $66,280 for December 2020. This is the first month-over-month decline for median household income in the U.S. since the coronavirus recession bottomed in August 2020.

The latest update to the chart tracking Median Household Income in the 21st Century shows the nominal (red) and inflation-adjusted (blue) trends for median household income in the United States from January 2000 through January 2021. The inflation-adjusted figures are presented in terms of constant January 2021 U.S. dollars.

The cause of the decline after months of recovery can be found in the renewed lockdown measures imposed by several state and local governments in late November or early December 2020, such as in California, Illinois, New Jersey, New York, and Pennsylvania, to identify the lockdown states with the largest populations. The data for December 2020 had indicated the rebound in median household income was slowing. The data for January 2021 confirms their negative impact.

Analyst's Notes

The BEA issued minor downward revisions of the received aggregate wage and salary dispersement data we utilize in generating our estimates of median household income for the period covering July 2020 through December 2020.

As promised earlier this year, we're rolling out a new near-real time economic indicator based on average individual earned income (income earned only from wages and salaries). Here's a sneak preview of what it looks like!

At first glance, average individual earned income per capita looks to be both more sensitive to changes in trends and more volatile than median household income. It's also easier and more direct to calculate. We'll discuss it at greater length after we've taken it for an extended test drive behind the scenes.

Other Analyst's Notes

Sentier Research suspended reporting its monthly Current Population Survey-based estimates of median household income, concluding their series with data for December 2019 before ceasing to operate in early 2020, as its principals have retired. In their absence, we are providing the estimates from our alternate methodology for estimating median household income on a monthly basis. Our data sources are presented in the following section.

References

Sentier Research. Household Income Trends: January 2000 through December 2019. [Excel Spreadsheet with Nominal Median Household Incomes for January 2000 through January 2013 courtesy of Doug Short]. [PDF Document]. Accessed 6 February 2020. [Note: We've converted all data to be in terms of current (nominal) U.S. dollars.]

U.S. Department of Labor Bureau of Labor Statistics. Consumer Price Index, All Urban Consumers - (CPI-U), U.S. City Average, All Items, 1982-84=100. [Online Database (via Federal Reserve Economic Data)]. Last Updated: 10 February 2021. Accessed: 13 January 2021.

U.S. Bureau of Economic Analysis. Table 2.6. Personal Income and Its Disposition, Monthly, Personal Income and Outlays, Not Seasonally Adjusted, Monthly, Middle of Month. Population. [Online Database (via Federal Reserve Economic Data)]. Last Updated: 26 February 2021. Accessed: 26 February 2021.

U.S. Bureau of Economic Analysis. Table 2.6. Personal Income and Its Disposition, Monthly, Personal Income and Outlays, Not Seasonally Adjusted, Monthly, Middle of Month. Compensation of Employees, Received: Wage and Salary Disbursements. [Online Database (via Federal Reserve Economic Data)]. Last Updated: 26 February 2021. Accessed: 26 February 2021.

There was exactly one big story driving the action for the S&P 500 (Index: SPX) in the final week of February 2021. Incoming data suggested the specter of inflation is not as distant as had been assumed. With its appearance, investors who had hedged their bond investments by buying stocks were shocked into selling to cover their losses as real interest rates for the 10-year U.S. Treasury rose above 1.5%.

For the S&P 500, although stock prices saw considerable volatility during the week, investors appeared to largely maintain their forward-looking focus on 2021-Q2, with the index' level remaining within the redzone forecast range we described in our previous update.

The new volatility does raise an interesting question for our redzone forecast. What would it mean if the level of the S&P 500 moved outside of the indicated range?

From our perspective, all that means is that circumstances have overridden the assumptions we made when we established it. As shown, it is based on the premise that investors would be focused on 2021-Q2 in setting current day stock prices in the early period covered by the redzone forecast, where they would transition to focusing on the more distant future quarter of 2021-Q4 sometime after March 2021.

But, a shock event like the bond market's 'tantrum without a taper' could compel investors to shift their focus to a different point of time in the future. At present, assuming no change in the expectations for future dividends, the worst case would be if investors shifted their focus to the current quarter of 2021-Q1, where they would be watching for the Federal Reserve's response to the bond market's 'tantrum' within this quarter. If that happened, stock prices could fall on the order of somewhere between 200-300 points. In the alternative futures chart, we would see the level of the S&P 500 drop below the redzone forecast range.

If investor expectations for future dividends change on top of that, then level to which the S&P 500 might go in that scenario would be affected to the extent those expectations change. We would see that change through the changing levels of the dividend futures-based model's alternative projections of where the S&P 500's level would be for when investors focus on particular points of time in the future.

The behavior of stock prices is complex, but not difficult to sort out if you have the right framework for understanding why they behave as they do. Regardless, the one thing that will determine how they behave in the immediate future in the random onset of new information. Speaking of which, here is our summary of the previous week's newsflow of market-moving headlines, which provides a good potion of the context needed to understand what new information investors were absorbing during the most volatile week for markets in months.

Monday, 22 February 2021

• Signs and portents for the U.S. economy:
• Oil jumps almost 4% as output slow to recover from Texas storms
• Fed minion expects strong recovery:
• Fed's Kaplan sees U.S. economy growing at least 5% this year
• Bigger stimulus developing in the Eurozone:
• EU to decide on extending suspension of Stability Pact in coming weeks: official tells paper
• Euro zone borrowing costs tumble on ECB Lagarde comments
• ECB minion swears there is no risk of lasting inflation in the Eurozone:
• No risk of overheating in euro zone - ECB's Villeroy de Galhau
• Nasdaq, S&P 500 end lower as U.S. yields rise; Disney lifts Dow
• Ten-year Treasury yield catching up to S&P 500 dividend
• Explainer: U.S. Treasury's cash drawdown - and why markets care

Tuesday, 23 February 2021

• Signs and portents for the U.S. economy:
• U.S. manufacturers grapple with steel shortages, soaring prices
• U.S. consumer confidence improves as COVID-19 cases fall; house prices accelerate
• Oil settles mixed amid post-storm uncertainty
• U.S. bill rates risk going negative, but stimulus could change the course
• Analysis: Bubbles, bubbles bound for trouble?
• Analysis: Prices lurch higher as Home Depot, other importers battle surging cargo, commodity costs
• Fed minion speaks to Congress, sees rapid growth once virus-fear driven lockdowns are lifted, not afraid of inflation:
• Fed Chair Powell's prepared remarks to Congress, Feb. 23, 2021
• Fed's Powell: 2021 GDP growth could be in range of 6%
• Powell says economy still needs Fed support, pushes back on inflation worries
• Powell's Econ 101: Jobs not inflation. And forget about the money supply
• Is Fed Chair Powell 'cool' with more fiscal aid? Suddenly he won't say
• Inflation also developing in the Eurozone:
• More expensive services lead euro zone inflation rebound in January
• S&P 500, Dow close higher in late session turnaround
• Stocks rebound on Powell policy remarks, oil gains

Wednesday, 24 February 2021

• Signs and portents for the U.S. economy:
• U.S. retail sales expected to jump on vaccine rollout, pent-up demand in positive sign for economy
• Oil rises after data shows slump in U.S. output amid Texas freeze
• U.S. new home sales blow past expectations in January
• Fed minions face negative rates, need Congress approval to launch digital currency, see need for endless monetary stimulus, but think future looks bright:
• Analysis: As bond selloff gets real, policymakers face fresh headache
• Fed's Powell: Digital dollar may require some congressional approval
• Data-focused Powell paves way for era of extended loose Fed policy
• Fed's Brainard says U.S. economy far from employment, inflation goals
• Fed's Clarida says U.S. economic prospects have brightened
• Mixed signs of growth, recession in Eurozone:
• Strong exports and construction boost German economy in fourth quarter
• German economy to shrink some 1.5% in early 2021: DIW
• BOJ minions becoming concerned over endless monetary stimulus:
• Former BOJ executive calls for 'genuine' review of central bank stimulus
• Wall Street advances as Fed's Powell soothes inflation worries

Thursday, 25 February 2021

• Signs and portents for the U.S. economy:
• Oil mixed, U.S. crude hits highest since 2019 as refineries restart
• U.S. pending home sales decline in January
• U.S. labor market showing tentative signs of improvement; manufacturing strong
• White House says it does not expect to issue budget proposal in February
• Fed minions super optimistic rapidly rising interest rates is a good thing, want to keep easy money policies:
• Atlanta Fed's Bostic says not worried about rise in bond yields with rates still comparatively low
• "Euphoria" over pandemic's end may make recovery particularly strong - Bostic
• Bostic: Recovery in "rough patch" but optimistic for faster growth
• Fed's George says rise in long-term rates a sign of optimism
• Bullard: Jump in bond yields proper response to stronger growth, inflation outlook
• Bullard says "rosy outlook" no cause to begin discussing pullback of Fed support
• Quarles cautions pandemic stimulus could obscure risks for banks
• NY Fed's Williams: GDP growth this year could be strongest in decades
• Bigger trouble developing in the Eurozone:
• Euro zone corporate lending growth slows as monthly flows dry up
• Positive signs of post-coronavirus recession recovery:
• World trade bounces back to pre-coronavirus levels: CPB monitor
• ECB minion commits to keep Eurozone interest rates low indefinitely:
• ECB will fight big rise in real rates: Schnabel
• Wall Street ends sharply lower, tech selloff weighs as bond yields climb
• Instant View: U.S. stocks tank as Treasury yields surge
• Bond markets left smarting from worst rout in years as reflation goes global

Friday, 26 February 2021

• Signs and portents for the U.S. economy:
• U.S. consumer spending rebounds in January
• U.S. House on verge of approving Biden's $1.9 Trillion COVID-19 Aid Plan
• U.S. Senate referee says Democrats cannot include $15 minimum wage in COVID bill
• Bigger trouble with inflation developing all over:
• Bond markets left smarting from worst rout in years as reflation goes global
• Analysis: Global bond rout turns up the heat on central banks
• Analysis: Big moves and liquidity woes in a U.S. bond 'tantrum without the taper'
• Column: Tantrum without the taper
• Analysis: Investors jumping the gun as TIPS, futures flag early Fed tightening
• As yields creep up, BOJ's Kuroda calls for 'stably low' rates
• Bank of England's Haldane warns inflation "tiger" is prowling
• Positive post-coronavirus recovery signs in India, China, mixed signs in Japan:
• India's economy returns to growth after shrinking for two quarters
• China's economy could grow 8-9% this year from low base in 2020: central bank adviser
• Japan's January factory output rises for first time in three months, retail sales drop
• ECB minions standing by, seem okay with rising interest rates, but thinking about unleashing more QE:
• ECB still in wait-and-see mode as yields rise
• ECB's Schnabel says rise in nominal yields may be welcome
• Exclusive: ECB's Stournaras calls for increasing bond buying to calm markets
• Nasdaq finishes higher, tech stocks retrace some losses
• Stock markets hit by bond whiplash

Meanwhile, Barry Ritholtz lists the positives and negatives he found in the past week's economics and markets news.