Hyperbolic and Other Geometries (Part II)

Note: This is part II of a post. For the first part, see Hyperbolic and Other Geometries (Part I).

Now that it is true that the Parallel Postulate is false, we can return to the constant C, which is the proportionality constant between the difference between 180 and the angles of a hyperbolic triangle and the area of that triangle. The constant C can differ for different hyperbolic planes. This is because not all hyperbolic planes have the same curvature. To fully explain curvature, one must also look at the Euclidean plane.

The Euclidean plane is defined to have curvature 0. That means that the surface of Euclidean geometry is flat. Hyperbolic planes, however, have negative curvature. There are different hyperbolic planes because there are different possible values for the curvature of the hyperbolic plane. This is unlike Euclidean geometry, which only has one possible value for the curvature, i.e. 0. But what happens if there is positive curvature? If there is positive curvature another whole type of geometry is introduced, which is even more radical than hyperbolic geometry. This is elliptic geometry. This is the only one of the three types of geometry that has a finite plane, which is one of a sphere. In elliptic geometry, all structures are projected onto a sphere. The theorems resulting include the fact that the angles of a triangle add up to more than 180. A representation of a triangle projected on a sphere is shown below.



This particular triangle is projected onto the Earth. The angles on the triangle add up to 230 degrees. Note that, in the close up image, the relative curvature of the area approximates zero, and the angles do add up to very close to 180 degrees. In elliptic geometry, Euclid's other postulates are violated. In this geometry, all but the fourth of Euclid's postulates are false (the fourth postulate is that all right angles are equal. The first postulate, that exactly one straight line can be drawn between any two points is false because, on a sphere, multiple lines can connect two points. Take longitude lines, for example. All longitude lines are drawn between the North and South poles, and in elliptical geometry, they all all straight lines. The second postulate, that a line can be continued indefinitely without ever intersecting itself is also false, because any latitude line on a globe eventually wraps its way around and intersects with itself. The third postulate (that a circle can be drawn with any center and radius) is false because if a circle is big enough, it will wrap around the sphere and intersect with itself, violating the very definition of a circle because the points on the edge of the circle would not all be the same distance from the center.

Just as the area of a triangle on a hyperbolic plane can be expressed with the equation 180-(x+y+z)=CΔ where C is a constant and Δ is the area of the triangle, the area of an elliptic triangle can also be expressed in terms of its angles, with the equation Δ=R^2(a+b+c-180). Again, Δ is the area of the triangle, while R is the radius of the sphere where the triangle is located. It is apparent that the quantity a+b+c-180 must be positive because the radius must be positive for the area of the triangle to be positive. If you combine these two equations and solve for C in terms of R, you obtain

C=-1/R^2

Going back to hyperbolic geometry, C, as the constant of variation for the area, must have a positive value. In hyperbolic geometry, therefore, for C to be positive, 1/R^2 must be negative (because then -1/R^2 would be positive). But this is impossible because all numbers square to a positive number or 0. Therefore, the solution to this equation for R must be imaginary, or the square root of a negative number. For information on imaginary numbers, see here. This seems impossible. How can a sphere have a radius that isn't positive or even a real number? It turns out an imaginary radius does have a geometrical meaning. The shape produced in this situation is called a pseudo-sphere. A representation of a pseudo-sphere is shown below.



In essence, this structure is kind of that of an inside-out sphere. The pseudo-sphere is wonderful, but what does it mean? It came about because the radius of a sphere that is a hyperbolic plane is a pseudo-sphere. It makes sense to say that hyperbolic geometry is actually projected onto this shape, just as an elliptic triangle is projected on a normal sphere. This may not seem like the case, but a pseudo-sphere is yet another Euclidean representation of hyperbolic geometry.

In conclusion, these are the only three types of geometry that fit the criteria. The criteria is that the geometries are homogenous and isotropic. Homogenous means that each point on the surface of the geometry is the same, and isotropic means that the perspective of the area around each point is the same as from any other point. There are other geometries that are combinations of the above three, and these usually include some regions with one geometry, and some regions with other types.

For the conclusion of this post, see Hyperbolic and Other Geometries (Part III)

Hyperbolic and Other Geometries (Part I)

Most people have some knowledge about regular plane, or Euclidean, geometry. The basis of many geometric theorems relies on five basic rules, or axioms, about Euclidean geometry that cannot be proved or disproved, but are simply guidelines that geometry must follow. People take these statements to be obviously true. The five are as follows (as stated by Euclid in Elements)

1. Exactly one straight line can be drawn between any two points
2. A finite line segment can be extended continuously without intersecting itself
3. A circle can be made from any center and radius
4. All right angles are equal to one another
5. Parallel Postulate: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if continued indefinitely, will eventually meet

In this post, the fifth of the five statements above is the one that will mainly be dealt with. This is the parallel postulate. In simple terms, as illustrated in the above image, if two lines are slightly inclined towards each other, in other words the angles between them and an intersecting line add up to less than 180 degrees, they will intersect eventually, whether it is a hundred units away, a million units away, or a googleplex units away, they will intersect. If alpha and beta in the picture above add up to more than 180 degrees, than the lines will incline away from each other and will meet only if you travel in the other direction. However, if the angles add up to exactly 180 degrees, the lines will be parallel. They will never meet. An equivalent property to this is the fact that all the angles of a triangle add up to 180 degrees. This is because if the lines are parallel, they never intersect. Yet another equivalent statement as the Parallel Postulate is the fact that for a line and a fixed point in a plane (that is a two-dimensional surface), only one line drawn through the point can be parallel to the given line (note that in three dimensions, two lines that never intersect aren't necessarily parallel. This is because they can be on two separate planes. Two lines with this relationship are called skew lines) All of the axioms above seem like common sense, right? However, most new branches of mathematics are opened up when one challenges the most simple axioms and properties of math.

In this case, we are challenging Euclid's axioms. The fifth of the statements, the Parallel Postulate, seemed the least obvious, and for centuries, various mathematicians attempted to prove this. Many of these mathematicians used a method known as proof by contradiction. Proof by contradiction begins by assuming what you're trying to prove is false and then trying to find a contradiction to a known axiom directly resulting from this fact. If a contradiction is reached, then the statement cannot be false and is therefore true. This method has be repeatedly used by mathematicians, and one notable example includes the proof that the square root of two is irrational (see here). The point is that many mathematicians assumed that the parallel postulate is false. If this is assumed to be false, then more than one line can be drawn through a point parallel to a given line. The many theorems that were derived from this assumption were very odd, indeed. However, no contradiction could be found. We now know that the Parallel postulate is neither true or false; it is true only in some cases. While searching for a contradiction, mathematicians found something much more interesting: a whole new type of geometry! In this geometry, a theorem that the angles of a triangle add up to less than 180 degrees can be derived. This new geometry is called hyperbolic geometry. It is very difficult to represent this geometry with images in Euclidean geometry, but one visual representation is shown below.



This is a representation of hyperbolic geometry in a Euclidean circle. In hyperbolic space, each angel or devil are precisely the same size. However, this representation has squeezed the entire plane of hyperbolic geometry into one circle, called the bounding circle. To the inhabitants of this Universe, you can go as far as you want, but you will never reach the bounding circle because the plane goes on forever. Shapes, lines, and other Euclidean structures can all be represented in hyperbolic geometry, although they may look different than in Euclidean space. Here is a triangle as it looks like on the hyperbolic plane.



The values shown are of each angle along with the total sum off to the right. Note the the sum of the angles, 107.1 degrees, is much smaller than what it "should" be, 180. In fact, it was discovered that the value of 180 minus the sum of the three angles is proportional to the area of the triangle, a relation somewhat unexpected by mathematicians. The following equation represents this relation.

180-(x+y+z)=CΔ

x, y, and z represent the three angles of a triangle and the symbol Δ represents the area of the triangle. In this equation, the constant C is not a set value for all hyperbolic planes. In fact, unlike Euclidean geometry, there are different types of hyperbolic planes, which will be mentioned later. Note that if the line segments on the above triangle were continued into lines, they would continue curving and also "reach" the bounding circle (obviously the finite curve that we see intersecting the bounding circle is indeed infinite from a hyperbolic standpoint).

Returning to the origin of this geometry, is the parallel postulate indeed false on this plane? Is is possible to draw more than one parallel line through a given point? Yes. The picture below is a different visual representation of hyperbolic geometry, but the structures are similar.



This representation may look different, but it is viewing the same plane. The lines in the lower right are parallel, but they curve in opposite directions, rather than remaining a fixed distance fro m each other. From this image, it is clear that infinitely many lines could be drawn through the same point as the given line, but would still never intersect it. Euclid's parallel postulate is indeed false in this geometry.

For the next part of this post, see Hyperbolic and Other Geometries (Part II).

2009 Season Summary

The 2009 Atlantic Hurricane Season was below average with eleven depressions forming, nine of which became tropical storms, three of which became hurricanes, and two out of the three became major hurricanes. In total, about $174 million in damages resulted (wealth value for 2009) and fifteen direct fatalities. No hurricanes hit the U.S. this year, and only one made landfall. Although this was a relatively quiet season, there were a few notable storms.

• Tropical Depression One forming in May made a tropical cyclone form in the Atlantic Ocean in May for three consecutive years
• No tropical storms formed until August 12, which was the latest start since 1992
• Hurricane Fred achieved major hurricane status the farthest south and east of any Atlantic cyclone ever recorded
• Tropical Storm Grace formed the farthest north and east of any tropical cyclone in the Atlantic Basin
• Hurricane Ida made landfall in the U.S. as a tropical storm in November, becoming one of only six cyclones to do so

Overall, due to an El Nino event, the 2009 Atlantic Hurricane Season was relatively quiet and not much damage resulted.

Intertropical Convergence Zone

The Intertropical Convergence Zone is an area near the Equator that circumnavigates the globe. Most of it lies between 10ºN and 10ºS. The trade winds, or the winds near the Equator, converge on this zone. Since the winds are rotated in different directions by the Coriolis Effect, no low pressure systems, and therefore no tropical cyclones, can form too near the Equator.



The Intertropical Convergence Zone over the East Pacific. Although no low pressure systems can form in the ITCZ, it is constantly marked by thunderstorm activity.

However, the boundary of the ITCZ, or the ITCZ Axis, is always shifting, and a tropical system can sometimes, but very rarely, from below 5ºN or above 5ºS. A notable example is Typhoon Vamei, which was a Western Pacific cyclone that reached tropical storm strength at a mere 1.5ºN in December of 2001, which is the closest formation to the Equator on record. Also notable is 2004's Cyclone Agni, which formed farther from the Equator than Vamei and moved towards it, eventually reaching the most southerly point of 0.7ºN before turning back northward. Claims on this system are disputed because it was not officially tracked until a few days after the record was set but a fair amount of evidence supports that Agni does indeed hold the record for closest cyclone to the Equator.



Cyclone Agni at record latitude, a mere 45 miles from the Equator.

Although the winds over the ITCZ hinder tropical cyclones from forming within the region, they do provide an important factor for tropical cyclone development: tropical waves. A vast majority of tropical cyclones form from tropical waves, which are areas of convection that typically move west over the ITCZ. After the wave moves northward, it then can develop a low pressure system, and eventually become a tropical system. Therefore, the ITCZ is an important factor in tropical cyclone development, even if tropical cyclones can't form in the area itself.



A map of the formation and progression of tropical waves before becoming tropical cyclones.

Hurricane Ida (2009)

Storm Active: November 4-10

On November 2, a low pressure system formed from a tropical wave along the Intertropical Convergence Zone. The low was situated in the extreme southwestern Caribbean, off the coast of Nicaragua. The low was nearly stationary for the next two days and it began to gather cloud cover and tropical characteristics. On November 4, Tropical Depression Eleven formed with 35 mph winds and a pressure of 1006 millibars. Later that afternoon, the winds near the center of circulation jumped to 60 mph and the system was named Tropical Storm Ida. Then, as it moved northwest, Ida reached a minimal hurricane strength of 75 mph winds and a 987 millibar pressure before making landfall in Nicaragua in the morning of November 5. As it encountered the mountainous terrain of the region, Ida promptly lost hurricane status, and it steadily weakened, becoming a tropical depression by late on November 5. During the morning of November 6, Ida crossed into Honduras, still maintaining tropical depression strength. Finally, later on November 6, Ida reemerged into the northwest Caribbean Sea. Ida regained tropical storm strength on November 7. During the day, the deep moisture and warm water of the region fed Ida, and it rapidly strengthened once again. By the evening of November 7, Ida was approaching hurricane strength, and was continuing north, towards the Gulf of Mexico. Ida then became a Category 2 hurricane with 100 mph winds and a pressure of 976 millibars. It brushed past the Yucatan Peninsula, causing tropical storm force winds and rain, but a majority of the wind was on the east side of the system due to a strong ridge of high pressure over the Bahamas. The pressure difference caused sustained winds of over 30 mph to rip through the Florida Keys and the surrounding region, despite the fact that the circulation was still fairly far away. In addition, rain from Ida extended farther to the southeast, reaching the Cayman Islands and beyond. Later on November 8, Ida reached its peak intensity of 105 mph winds and a pressure of 976 millibars. Then as Ida entered the northern Gulf of Mexico, it encountered cooler water, and weakened once again. By the afternoon of November 9, Ida had become a tropical storm once again, and was already battering the coasts of Alabama, Louisiana and the Florida Panhandle with high surf and tropical storm force winds and rain. Overnight, Ida weakened further and began its extratropical transition. By the morning of November 10, the center of Ida was sill offshore, but all convection associated with the system had moved to the north over much of the southeast U.S., producing rain and thunderstorms, including up to six inches of rain locally in parts of Florida. As Ida made landfall in Alabama, it promptly weakened to a tropical depression and became extratropical. It then combined with a westward moving cold front and another low pressure system approaching North Carolina from the east to bring a huge rain and wind event to the entire U.S. east coast. In addition, a very strong high pressure system was situated over Maine (it was a powerful one, with a pressure exceeding 1035 millibars) causing east-to-west wind to bring surf and tropical storm force gusts to much of the coast. The high pressure also blocked the system, and it moved very slowly. Rain and wind continued for the next three days and some areas accumulated over 10 inches of rain. By November 13, the low pressure system had moved up to the coast of New Jersey and was finally weakening. The system sped off to the northeast and left the U.S. on November 14. Ida was a notable storm in El Salvador because it contributed to a mudslide that killed 124 people, but Ida also became a powerful and dangerous nor'easter after becoming extratropical, killing an additional 10 people. Ida also caused $2.15 million in damage.



Ida strengthening as it enters the Gulf of Mexico.



Track of Ida.

Tropical Storm Henri (2009)

Storm Active: October 6-8

Near the end of September, a tropical wave emerged off the coast of Africa and moved westward. By October 1, the system already was associated with a large area of showers and thunderstorms. The deep tropical moisture enriched the system and contributed to its organization, but it did not develop a well-defined center. The wave moved northwest, out of favorable conditions, but convection persisted. Then, on October 6, a rapid intensification occurred, and the system was declared Tropical Storm Henri. Henri's convection was displaced to the east of the center by El Nino-related west to east sheer, much like Danny and Erika before it. Despite these adverse conditions, Henri gained strength, reaching its peak intensity of 50 mph winds and a pressure of 1005 millibars. Henri paralleled the northern islands of the Caribbean Sea to their north, causing raised surf. Henri weakened later on October 7, and, overnight, became a tropical depression. By the early evening of October 8, Henri had disintegrated into almost nothing on satellite imagery, and it decayed into a remnant low. The low quickly dissipated on October 9. No significant effects resulted from this system.



Henri at peak intensity north of the Caribbean Sea.



Track of Henri.

Tropical Storm Grace (2009)

Storm Active: October 4-5

On October 1, the center of a non-tropical low over the Azores developed a small area of showers. However, the showers didn't persist and no tropical cyclone formed. The low moved slowly northeast, hindered by a stationary front to its north into October 4. Then, an area of convection flared up rapidly within the low that evening, resulting in it being classified as Tropical Storm Grace. It formed at 41.2 N and 20.3 W, making it the farthest northeast a tropical cyclone has ever formed on record, Since the low was already fairly strong (995 millibars before Grace's formation) Grace already had reached a strong tropical storm intensity of 65 mph winds and a pressure of 990 millibars. Grace's movement quickened to 25 mph, as it went speeding off to the northeast. Despite being in a hostile area for tropical cyclone development, Grace strengthened into the morning of October 5, reaching its peak intensity of 70 mph winds and a pressure of 989 millibars. At its peak intensity, Grace was a very small system, less than 100 miles across. In comparison, the widely scattered shower activity of the extratropical low spanned hundreds or even a thousand miles. Throughout the day of October 5, Grace continued northeast, reaching a forward speed of 30 mph, and began to weaken. It was then absorbed by a frontal boundary late on October 5. The combined system caused scattered showers over the British Isles over the next day but no damage resulted.



Grace at its peak of a small, intense tropical storm in the extreme northeast Atlantic.



Grace's track.

Tropical Depression Eight (2009)

Storm Active: September 25-26

A tropical wave moved off of Africa on August 23, and began to develop disorganized showers and thunderstorms as it moved to the west-north-west at about 15 mph. On September 24, the Cape Verde Islands received scattered showers and thunderstorms as it passed to the south of the region. On September 25, the wave began to organize west of the Cape Verde Islands, despite being in an area only marginally favorable for tropical cyclone formation and started to develop a center of circulation. The system was declared Tropical Depression Eight at 5 p.m. on September 25 already at its peak intensity of 35 mph winds and a pressure of 1008 millibars. Over the next day, the depression moved northwest into cooler waters and became less organized. Then, exactly a day after formation, Eight lost its center of circulation and dissipated. The tropical wave continued to produce shower activity as it moved northwest and was monitored for regeneration, but no new low formed, and the possibility for reformation was gone by September 27.



The tropical wave destined to become Tropical Depression Eight organizing in the East Atlantic.



Track of Eight

Hurricane Fred (2009)

Storm Active: September 7-12

On September 6, a strong, organized, tropical wave associated with an area of low pressure moved off of Africa and quickly developed into Tropical Depression Seven the next day. The depression passed south of the Cape Verde Islands and intensified into Tropical Storm Fred on September 7. Fred's center became well-defined, and the system strengthened through the next day, becoming Hurricane Fred on September 8 with 75 mph winds and a pressure of 987 millibars. Hurricane Fred jumped to Category 3 intensity in just 12 hours and reached its peak intensity of 120 mph winds and a pressure of 958 millibars on September 9. At his time, Fred had a distinct eye feature, and it became the most southeasterly major hurricane ever to form in the Atlantic Ocean. Fred curved to the north, due to a ridge to its west, and its southeast side weakened. As a result, strong wind sheer ripped at the system from that direction, causing Fred to weaken, becoming a tropical storm on September 11. Convection began to separate from the center to the north, and Fred became nearly stationary, drifting east, then south, and weakening all the time and making a tight loop in the Eastern Atlantic. By mid-afternoon on September 12, Fred was a minimal tropical storm, with almost no convection. It was then downgraded to a remnant low that night, and moved west. The low continued to produce shower activity, but dissipated on September 16. However, a new low formed in association with Fred's remnants 500 miles south of Bermuda and the area was monitored once again. The low continued to drift west, and finally dissipated, becoming part of a frontal boundary, on September 20.



Hurricane Fred at Category 3 intensity in the far east Atlantic.



Track of Fred. Note that the triangles signifying Fred's remnant low continued much farther west before dissipating.

Tropical Storm Erika (2009)

Storm Active: September 1-3

On August 26, a tropical wave moving off of Africa became associated with a broad area of low pressure south of the Cape Verde Islands, producing shower and thunderstorm activity. However, the cloud cover and organization diminished over the next few days as the system moved west. By August 31, though, the low had regenerated into an organized ball of convection. The system was organized enough to be a tropical storm and had tropical storm force winds, but i still wasn't classified. This was because it lacked a center of circulation. The low was an elongated oval, widest north to south, and a center could have formed anywhere along the oval. During the afternoon of September 1, a center began to for, evident on visible imagery as a point just east of the cloud cover associated with with the system. Therefore, like the previous storm Danny, the system skipped tropical depression status and was upgraded directly to Tropical Storm Erika, with 50 mph winds and a pressure of 1007 millibars. Later that night, the pressure dropped to 1004 millibars and the winds increased to 60 mph, but Erika took a southerly turn and weakened the next morning. By mid-morning, Erika began to interact with the northern Leeward islands of the Caribbean Sea, causing wind and rain. Later on September 2, Erika entered the Caribbean, still maintaining minimal tropical storm winds of 40 mph. Erika struggled westward, its center well west of the convection, similar to Tropical Storm Danny's appearance several days earlier. Erika meandered through the Caribbean for another day, before weakening to a tropical depression south of Puerto Rico on September 3. Overnight, Erika weakened into a remnant low. The next day, the low dissipated. Damage was minimal and no fatalities resulted from Erika.



Erika near peak intensity. Even then, Erika was very disorganized.



Track of Erika.