2010 Season Summary

The 2010 hurricane season was well above average, with

21 cyclones attaining tropical depression status
19 achieving tropical storm status
12 hurricanes
and 5 major hurricanes

This is higher than my predictions of

18 cyclones attaining tropical depression status
17 cyclones attaining tropical storm status
7 cyclones attaining hurricane status
4 cyclones attaining major hurricane status

particularly in the hurricanes category.

This activity (19 named storms) was tied for the third most ever recorded in the Atlantic basin. The most powerful cyclone of the season was Igor, which attained a peak intensity of 155 mph winds and a minimum pressure of 925 mb. It also was the largest tropical cyclone ever to form in the Atlantic basin in terms of tropical storm wind diameter. Due to its colossal nature, Igor was also the third wettest tropical cyclone every recorded in Canada, dumping 9.37 inches of rain in one location in Newfoundland.

Also:

• 8 storms formed in September (tied for a record high)
• Four cyclones (Alex, Karl, Matthew, and Richard) made landfall in Belize, although two of them at tropical depression status (record high)
• Two Category 4 hurricanes (Igor and Julia) existed simultaneously for a brief period of time, an occurrence that has not happened since 1926

Overall, the 2010 Atlantic hurricane season was a very active one, and impacts were mostly in the Caribbean and Central America. The United States, surprisingly, was barely affected, with no hurricane landfalls.

Hurricane Tomas (2010)

Storm Active: October 29-November 7
On October 25, a tropical wave formed in the extreme southeastern Atlantic Ocean, near 5ºN. The wave produced only scattered shower and thunderstorm activity as it moved west over the next few days. It was a vigorous tropical wave, however, and it developed a low pressure center on October 27. The low adopted a general northwest motion, and deepened significantly over the next two days. By the afternoon of October 29, the system had a very organized circulation and outflow, and the confirmation of a closed low at its center merited the upgrading of the system into Tropical Storm Tomas.

Tropical Storm Tomas was already in a state of rapid intensification, and the winds increased rapidly as the cyclone approached the Caribbean Islands, moving westnorthwest between 10 and 15 mph. During the morning of October 30, Tomas passed directly over Barbados, with peak winds of 70 mph, causing fairly significant damage. Tomas developed a very wide eye feature (about 40 miles across) just before noon on October 30, and it was then organized enough to be upgraded to a Category 1 hurricane.

The system promptly made a direct landfall in St. Vincent during that afternoon, and heavy rain and tropical storm winds affected islands up to 100 miles north and south along the Windward and Leeward Islands. The wide eye clouded over with a flare of convection, and Tomas continued to strengthen, becoming a Category 2 by later that night. However, southwesterly shear and dry air began to impact the west side of the system early on October 31, weakening it to a Category 1 storm by the afternoon. The center became ragged in appearance, and lost definition as a result of harsh atmospheric conditions.

Tomas weakened further into a tropical storm during the night, and only stabilized on November 1, when the winds dropped to 45 mph. Tomas was pushed on a general westsouthwest course during the day. Tomas's intensity fluctuated with large variations in convection over the day of November 2. The cyclone's forward speed also decreased as it reached the edge of a ridge to its north and steering currents weakened. For a brief period on November 3, Tomas degenerated into a wide area of scattered convection covering the entire southwest Caribbean, and was therefore downgraded to a depression, but the conditions for development drastically improved later in the day and Tomas turned towards the north, and the storm underwent a fast strengthening process, regaining tropical storm status. Tomas reached an intensity of 50 mph winds, and maintained it for the next day as it slowly moved northward. Wide rain bands began to sweep across Jamaica, Haiti and Cuba by the afternoon of November 4. As Tomas approached land, it rapidly strengthened into a hurricane, reaching its secondary peak intensity of 85 mph winds and a pressure of 984 mb as it passed just west of Haiti on November 5.

Cuba and Haiti both experienced tropical storm conditions, as well as hurricane force in some areas of Haiti, as the day went on, and as Tomas began to accelerate northeast, it interacted with the land around it briefly, and weakened back to a minimal Category 1 hurricane later in the evening. Tomas passed over the Turks and Caicos islands overnight, but emerged over open Atlantic waters on November 6, weakening back to tropical storm. Unexpectedly, Tomas once again regained hurricane strength late on November 6, but a cold front quickly overtook the system, and Tomas rapidly transitioned into an extratropical low on November 7. 41 fatalities and $572 million in damage directly resulted from Tomas in the Caribbean Islands, but Tomas is also indirectly linked to an epidemic of cholera in Haiti.



Hurricane Tomas intensifying as it enters the Caribbean. A fair amount of wind shear is evident on the south side of the system.



Track of Tomas.

Hurricane Shary (2010)

Storm Active: October 28-30
A trough of low pressure formed in the Caribbean on October 26. The trough was associated with an area of convection, but strong upper-level winds prevented development. However, on October 28, the shear relaxed enough for a central low pressure to form. However, the convection remained disassociated with this low until late on October 28, when the system developed an eyewall. At this point, the low was upgraded to Tropical Storm Shary.

Tropical Storm Shary sped to the northwest through the night, and shear began to increase on the system once again, giving it a lopsided appearance. This shear was produced by a strong front moving east off of the U.S. and this front also began to turn Shary to the north and then northeast. Despite adverse conditions, Shary strengthened as it began the turn, intensifying to 60 mph winds and a pressure of 1000 mb during the afternoon of October 29. The cyclone then made its closest approach to Bermuda, causing only showers and gusty winds however, as it passed well to the east. Shary's circulation was largely exposed through the coming day, but it continued to strengthen, becoming a minimal hurricane early on October 30.

By this time, Shary was speeding off to the northeast, and it briefly reached its peak intensity of 75 mph winds and a pressure of 989 mb before quickly becoming extratropical and being absorbed by a front later that afternoon. No damage resulted from Shary.



Hurricane Shary at peak intensity.



Track of Shary.

Hurricane Richard (2010)

Storm Active: October 20-26
On October 16, an area of showers and thunderstorms developed in the extreme southwestern Caribbean. The area drifted to the northwest, and interaction with the Nicaragua-Honduras area inhibited development for a time on October 18 and 19. However, after emerging over open water, the circulation improved, and the system moved slowly northeast. Late on October 20, the system became organized enough to be Tropical Depression Nineteen. By this time, steering currents had weakened, and the cyclone had reverted to a slow southeast movement. Dry air existed near the circulation over the next day, but intensification occurred nonetheless, and Nineteen became Tropical Storm Richard on October 21.

The system did not intensify for almost a day, but convection increased as dry air moved away from the system and a ridge built over the Gulf of Mexico, steering Richard back to the west by October 22. The system finally began to strengthen that day, rapidly intensifying into a strong tropical storm the next morning. Due to its proximity to Honduras, tropical storm conditions began for coastal areas by late on October 22. Richard began accelerating to the westnorthwest late on October 23, and a burst of convection the following morning caused Richard to intensify rapidly, becoming a category 1 hurricane.

The cyclone continued to intensify and made landfall in Belize at its peak strength of 90 mph winds and a minimum pressure of 981 mb during the evening of October 24. It quickly weakened over the next day, becoming a tropical depression by October 25. The system reemerged over the Gulf of Mexico early on October 26, but conditions were hostile for restrengthening and Richard swiftly weakened to a remnant low. The effects of Richard were $24.7 million in damage and 2 fatalities.



Richard as a Category 1 hurricane before landfall.



Track of Richard.

Hurricane Paula (2010)

Storm Active: October 11-15
On October 7, a broad area of low pressure developed in the southwestern Caribbean. Disorganized showers and thunderstorms remained associated with the system as it drifted generally to the northwest over the coming days. A low pressure center formed on October 9, and deepened thereafter, becoming a tropical depression during the morning of October 11, although not being formally recognized as a tropical system until later that afternoon. By that time, the cyclone was already a strong tropical storm, and was named Paula.

The system was in the midst of rapid intensification, and was a hurricane by the morning of October 12. It turned more to the northnorthwest over the next day, but continued to strengthen, exploding into a Category 2 (albeit a small one) by the afternoon of that same day, as it approached the Yucatan Peninsula. It stalled just offshore to the east later on October 12, still maintaining its peak intensity of 100 mph winds and a pressure of 981 mb. Since Paula was a small storm, only minimal rain and wind affected the Yucatan itself, and a jet stream just to the north of the system started to push Paula to the east and weaken it by the afternoon of October 13. The system accelerated eastward slightly, and made landfall in western Cuba on October 14, as it weakened to a tropical storm. Paula continued to degenerate, becoming a remnant low by October 15. It dissipated the next day. Paula was a very small storm, and damage was therefore limited, with only one fatality recorded.



Paula at peak intensity. The system remains very small, with a correspondingly small eye feature.



Track of Paula.

Hurricane Otto (2010)

Storm Active: October 6-10
On September 28, a tropical wave over the Central Atlantic began to produce an area of showers and thunderstorms. The next day, another tropical wave to its east also began to be monitored for development. The two systems moved west, but the second caught up with the first and the two waves combined on September 30. The combined disturbance produced a wide area of showers and thunderstorms as it moved westnorthwest, but wind shear increased, and the system remained disorganized. A low pressure center began to form in association with the system, and the low deepened as it passed over the Leeward Islands on October 3-5. By October 6, a surface circulation had formed. However, unlike a tropical cyclone, the center of the system had an upper-level low situated above it, rather than an upper level high, and this fact, combined with the limited convection that was only prevalent on the southeast side of the center, resulted in the classification of the system as Subtropical Depression Seventeen early on October 6. Seventeen's convection wrapped around the center the next day, and the winds reached gale force that evening, meriting the naming of the system as Subtropical Storm Otto.

By late on October 6, Otto's winds had rapidly increased, and the cyclone had reached an intensity of 65 mph winds and a pressure of 990 mb. However, the convection remained very sparse throughout the night, and intensity was difficult to judge, although the ragged appearance of the circulation suggested a slight weakening during the morning of October 7. By later in the morning, the upper-level low that had been shearing the circulation weakened, the core had warmed, and a distinct central eyewall had appeared as the system turned northeast. Otto was now a tropical cyclone, and it was officially classified as such at 11:00 am EDT on October 7. Otto's cloud cover continued to increase as it accelerated eastnortheast, and the system strengthened further, becoming a hurricane by October 8. The system reached its peak intensity of 85 mph and a minimum central pressure of 972 mb, before beginning to weaken. The system picked up speed as it moved out to sea, and it became a tropical storm late on October 9. Otto lost most of its central convection and was displaced to the north over the next 12 hours. As a result, the system was extratropical by midmorning on October 10. The cyclone subsequently impacted the Azores with some rain and wind as it weakened, dissipating on October 12.

Otto caused $20 million in damage but no deaths were reported, most damage being caused by flooding in the Caribbean Islands when the cyclone loitered to the north. As much as 17 inches of rain was reported in parts of Puerto Rico over a six day period from October 3-8 (this and similar rainfall reports courtesy of the Hydrometeorological Prediction Center).



Hurricane Otto at peak intensity speeding off into the open Atlantic.



Track of Otto.

Tropical Storm Nicole (2010)

Storm Active: September 28-29
During the final week of September, Tropical Storm Matthew dissipated over Mexico. However, the huge amount of moisture left over from this system was accompanying a broad area of low pressure over Central America and the western Caribbean. Disorganized showers and thunderstorms began to appear in this area during the day of September 26, and pressures dropped over the area over the coming days. On September 28, a closed center formed, and the system was upgraded to Tropical Depression Sixteen. The depression's center was broad, with the only area of convection to the southeast of the circulation, but a band formed to the northeast of the center later that day, as well as the center itself becoming slightly more defined, and these factors resulted in the pressure dropping slightly as the system moved northnortheast, but not a promotion to tropical storm status. The depression made landfall in Cuba, but as it was a very asymmetrical and broad cyclone, it was very difficult to classify one way or the other. However, the presence of tropical storm force winds near the center was enough to push the cyclone to Tropical Storm Nicole during the morning of September 29, while still over Cuba.

The system emerged over water but the huge circulation lost the little tropical characteristics that it had, and was declared dissipated later that day, shortly after reaching its peak intensity of 40 mph winds and a pressure of 996 mb. However, the rainfall was by no means over. The remnant moisture of Nicole combined with an extratropical low off of North Carolina and a stationary front over the northeast to bring torrential rainfall to the region from Maine to Florida, with local amounts exceeding fifteen inches. This storm activity finally ceased by October 1. This cyclone caused 13 fatalities and $151.9 million in damage, but this does not include additional damage wreaked by the combined system that impacted the northeast U.S.



Nicole as an odd-looking tropical storm near Cuba.



Track of Nicole.

Tropical Storm Matthew (2010)

Storm Active: September 23-26
On September 19, a trough of low pressure formed in the south central Atlantic. It slowly organized organization and a low pressure center appeared on September 21. The system gained convection, and the convection showed signs of a organized circulation by September 23, meriting the promotion of the low to Tropical Depression Fifteen. The tropical depression's center was fairly open until later that day, when a band circumnavigated the center for the first time, and the system was named Tropical Storm Matthew.

Matthew moved quickly westward through the southern Caribbean, and strengthened slowly over the next day. However, Matthew only reached an intensity of 50 mph winds and a pressure of 998 mb, before making landfall in northern Nicaragua during the afternoon of September 24. It maintained an intensity of 50 mph over land, and briefly emerged over the Gulf of Honduras on September 25, before beginning to weaken over Belize later that day.

The system quickly weakened to a tropical depression, and was a remnant low by early on September 26. However, the large area of moisture associated with the large circulation of Matthew caused continual heavy rain and flooding through the next few days. Matthew was the direct cause of several mudslides in Central America, causing 109 fatalities.



Matthew at its peak intensity before making landfall in Mexico.



Track of Matthew.

Hurricane Lisa (2010)

Storm Active: September 20-26
The tropical wave that became Lisa emerged off of Africa on September 16. The cloud cover associated with the wave remained disorganized for a day, but a broad low pressure center formed on September 17. The low took a turn northwest before slowing its motion and drifting to the north over the next few days. This uncertain motion was caused by a weakness in the usual Azores high that pushes cyclones to the west, a weakness that was enlarged by Hurricane Julia a few days earlier. Meanwhile, the system became more organized, and was declared a tropical depression late on September 20. The depression soon strengthened into Tropical Storm Lisa.

The cyclone meandered northeast, then east, then south, and then east again through the next day, with no change in intensity. Some convection was lost during the day of September 21, and the system weakened to a tropical depression, still maintaining a slow east motion. By September 23, convection had organized enough for Lisa to become a tropical storm again, and the system was nearly stationary during that day. However, Lisa adopted a northward motion during the day of September 24, and continued strengthening.

By late on September 24, Lisa had quickly intensified to a hurricane, reaching its peak intensity of 80 mph winds and a pressure of 987 mb just before midnight. An eye feature even made a brief appearance. However, Lisa began quickly weakening the next day, as it moved into cooler waters. The system was a remnant low by the afternoon of September 26. Lisa moved north and quickly dissipated. The cyclone affected no land masses.



Lisa as a hurricane. It is obviously a very small storm.



Track of Lisa.

Hurricane Karl (2010)

Storm Active: September 14-18
Late on September 9, a low pressure system formed to the east of the northeastern coast of South America. The low stalled near the Windward Islands, and crossed into the Caribbean on September 10. The low produced a lot of disorganized shower and thunderstorm activity which brought stormy conditions to much of the Caribbean over the next few days as it moved west, but little of this convection showed evidence of any circulation. However, on September 14, a closed circulation became evident from an aircraft reconnaissance mission sent to investigate the system, and a small pocket of tropical storm force winds allowed the low to skip depression status and become Tropical Storm Karl.

Karl lost most of its convection during the night, but still strengthened slightly, before a burst of convection near the center caused the cyclone to intensify to 65 mph winds early on September 15 just before landfall in the southeast Yucatan Peninsula. Since Karl made landfall near the Mexican border, portions of both Mexico and Belize were struck by tropical storm force conditions. Karl weakened over land to a minimal tropical storm, but reemerged over the Bay of Campache by early on September 16. Karl immediately began a strengthening trend over water, and the system quickly intensified into a hurricane later on September 16. An eye feature developed over the next day, and Karl rapidly intensified into a major hurricane, reaching its major hurricane peak intensity if 120 mph winds and a pressure of 956 mb on early on September 17, before weakening slightly just before landfall in Veracruz, Mexico with maximum winds of 115 mph.

Karl quickly began to weaken over land, losing its major hurricane status quickly later on September 17. Karl was ripped apart by the mountainous regions of central Mexico as it moved southwest inland, but these same mountainous regions caused a large amount of flooding over the affected area. 22 fatalities and $3.9 billion in damage are direct effects of the cyclone.



Hurricane Karl before striking the Mexican coast.



Track of Karl.

Hurricane Julia (2010)

Storm Active: September 12-20
Early in the second week of September, a strong tropical wave was already very evident over Africa, and it was monitored for development even before leaving the coast. By the time it did emerge over the Atlantic Ocean on September 11, it was very organized and was already showing tropical characteristics. As a result, the system was declared Tropical Depression Twelve on September 12. As the depression turned westnorthwest, it intensified into Tropical Storm Julia late on September 12.

Early on September 13, the southernmost Cape Verde islands experienced tropical storm conditions, as Julia passed just to the southwest. Julia took a more northward track than the cyclones before it, but it still intensified over the next day, as the predecessor of an eyewall formed near Julia's center, signifying a very healthy cyclone. The storm continued this strengthening trend, and became a Category 1 hurricane early on September 14.

It continued to strengthen through the morning as a structure that was almost an eye appeared, but Julia stabilized later in the afternoon after strengthening rapidly to its peak intensity as a Category 4 hurricane with 135 mph winds and a pressure of 950 mb early on September 15.

The cyclone turned farther to the north and began a general northwest motion. It began to encounter less favorable conditions as it approached cooler water and shear associated with the outflow of Igor. However, during its time as a Category 4, Julia set the record for strongest Atlantic cyclone east of 35ºW, surpassing the record set by Hurricane Fred just a year earlier. Julia continued to weaken over the next day, but wind shear died down slightly later on September 16, as Julia took a more westward turn. Julia maintained its Category 1 intensity over the next day, despite entering the outflow of the much larger and powerful Igor.

Julia continued weakening, and the center became separated from the convection on September 17. As a result, Julia soon became a tropical storm. Julia continued weakening into September 19, as it turned north and then northeast, accelerating over the open ocean. The system became extratropical on September 20, and started to be absorbed over the next day as a frontal boundary associated with Igor engulfed it. Julia caused only minimal damage while passing the Cape Verde islands, and was only notable for being a major hurricane very far east.



Julia near peak intensity.



Track of Julia.

Hurricane Igor (2010)

Storm Active: September 8-21
On September 6, a strong tropical wave emerged off of Africa. A low became associated with the system on September 7, and the system continued to organize, despite significant shear. On September 8, the system was organized enough to skip the tropical depression stage and intensify directly into Tropical Storm Igor.

The convection and circulation of Igor were at a very high level of organization, even with two low pressure systems in the vicinity. These three lows shared a low pressure trough extending from just off Africa to a few hundred miles north and west, but Igor began to strengthen later on September 8, and asserted its dominance. Igor meandered slowly to the west, before taking a turn north and then northwest on September 9. As it moved into greater shear, it weakened, becoming a tropical depression later that day, as the center became exposed from the east side. Despite this, the circulation deepened, the convection became more organized, and the system regained tropical storm status the next day. The shear had abated once again and Igor began to strengthen. During the morning of September 11, Igor developed an interesting eye feature on the north side of the system, and the storm approached hurricane strength. However, convection decreased during the afternoon before a burst of convection during the evening caused Igor to become a hurricane.

An eye became evident within this convection early on September 12, and Igor underwent explosive strengthening, ballooning from a minimal hurricane to a amazing Category 4 by that afternoon as a result of a drop of 50 mb in 12 hours. Igor's forward motion slowed and the system moved due west throughout that same day. Igor continued to strengthen, reaching an intensity of 150 mph late on September 12, and this remarkable intensity was maintained throughout the day of September 13, as a beautiful symmetric eye dominated the system. By that evening, surf began to increase in the Northern Leeward Islands as Igor approached from the west. Igor made a slight westnorthwest turn that night, and an eye replacement cycle destabilized the system, weakening it. However, it remained a Category 4 through the day of September 14, before organizing further and strengthening once again during the evening. Igor reached a peak intensity of 155 mph and a pressure of 925 mb later that night before the eye clouded over and a weakening trend commenced.

However, Igor organized once again, and the fluctuations in intensity continued as Igor became a powerful Category 4 once more with 145 mph winds. The convection became slightly asymmetrical, with the bulk of the cloud cover on the north side on September 16, but the moisture evened out during the evening as a a pronounced rain band formed south of the center. Also, during the day of September 16, Igor's tropical storm wind field broadened to 506 miles in diameter, making it the third-largest Atlantic hurricane on record. Igor began weakening, however, and finally lost it's Category 4 status during the late afternoon of September 16, after maintaining it for four days. The system continued to weaken over the next day. By the afternoon of September 18, Igor was a Category 2, but the windfield was still broadening, and squally weather was already sweeping over Bermuda. By later that night, Igor's tropical storm force windfield engulfed Bermuda, and the winds increased throughout the day, despite the fact that Igor weakened to a Category 1.

The center of Igor passed just to the west of Bermuda late on September 19, and the island saw sustained winds near hurricane-force as a result. Igor accelerated northeastward, and maintained a minimal hurricane status while the extratropical transition began on September 20. However, this transition wasn't completed by the time Igor passed Newfoundland, and the center passed just to the east of the island, causing sustained winds near hurricane force and dumping over 9 inches of rain in some areas, causing flooding. Igor finally became extratropical on September 21. Igor caused 3 fatalities and about $100 million in damage. The cyclone was also notable for being the largest Atlantic hurricane ever recorded, with a tropical storm force wind diameter of 920 miles, until it was surpassed by Hurricane Sandy of 2012.



Igor near peak intensity.



Track of Igor.

Tropical Storm Hermine (2010)

Storm Active: September 5-7
On September 3, a low pressure system formed in the eastern Pacific and quickly became Tropical Depression Eleven-E. However, the system made landfall in Mexico the next day without becoming a tropical storm. The remnant low of the system moved into the Bay of Campeche on September 5 and began to organize again. The low was associated with a very large area of showers and thunderstorms covering a significant portion of the western Gulf of Mexico. Late on September 5, the system was organized enough to be upgraded to Tropical Depression Ten. Ten quickly strengthened, and became Tropical Storm Hermine early on September 6.

Hermine quickly organized, and began rapidly strengthened as it moved generally to the north. Hermine reached its peak intensity of 65 mph winds and a minimum pressure of 991 before making landfall in extreme north Mexico late on September 6. Hermine crossed the U.S.-Mexico border inland a few hours later, as it steadily weakened. Hermine maintained minimal tropical storm status for a fairly long time inland, but finally weakened to a tropical depression by the evening of September 7. By later that night, it was no longer monitored by the national hurricane center, but it still maintained tropical depression status, as it tracked northward through the central U.S. The depression merged with a frontal boundary on September 9. The remnant moisture combined with the frontal system caused heavy rain from the midwest to the northeast over the next couple of days, before moving off the coast on September 12.



Hermine inland over Texas, still maintaining tropical storm strength and a healthy outflow.



Track of Hermine.

Tropical Storm Gaston (2010)

Storm Active: September 1-2
A tropical wave emerged off of Africa on August 29, and immediately began to organize, developing a low pressure center rapidly. By the beginning of September, a defined center had formed, and the system became Tropical Depression Nine early on September 1. The depression quickly crossed the border to tropical storm strength and was named Tropical Storm Gaston.

However, some Saharan dry air was still embedded in the system, preventing deep convection in the center. Meanwhile, the cyclone was tracking only very slowly westward, due to the presence of a trough to its north. The dry air present in the system weakened Gaston to a tropical depression and then a remnant low by the afternoon of September 2. However, early on September 3, convection associated with the remnant increased, and organization continued to increase over the next few days. Despite this, the low lost its good circulation, and, although convection persisted, the chance of development was significantly decreased by September 7 as it passed through the Caribbean. On September 8, the low dissipated. Gaston affected no landmasses and therefore had no impact.

Note: It is believed that the remnants of Gaston may have briefly attained tropical depression status again on September 4, but post-season analysis will confirm this after the conclusion of 2010.



Gaston on September 4, possibly a tropical depression.



The track of Gaston, with appropriate changes made from post-storm analysis.

Tropical Storm Fiona (2010)

Storm Active: August 30-September 3
On August 26, a strong tropical wave emerged off the coast of Africa, adopting the same general track of Danielle and Earl before it. By later that same day, a low pressure center became embedded in the wave. Although the outflow and circulation of the system was very organized from the beginning, convection remained minimal and a tropical depression didn't form during the next few days. On August 28, a burst of convection appeared at the center, but it ebbed away over the next day. By August 30, the system was producing tropical storm force winds, and a center was found, causing the low to be promoted to Tropical Storm Fiona, with no intermediate tropical depression stage.

Its initial intensity was 40 mph winds and a pressure of 1007 mb. Fiona sped off to the west and westnorthwest over the next day, and tropical storm watches were issued for some of the Northern Leeward Islands in preparation for possible tropical storm conditions in areas that were still recovering from Hurricane Earl. Fiona's motion was at least 20 mph until September 1, when the presence of Earl to its east slowed its motion. The outflow of Earl and Fiona kept a certain distance between the two, and Fiona actually became more organized as it slowed down, unexpectedly strengthening into a strong tropical storm by the morning of September 1, with winds of 60 mph. However, later on September 1, Fiona peaked at 60 mph and a pressure of 997 mb, before losing most of its cloud cover and weakening. Meanwhile, it turned to the northwest and sped up again through the morning of September 2. Fiona recovered some convection during the day, but intense shear exposed Fiona's circulation, and as it turned north, it continued to weaken. As Fiona struggled north-northeast, its pressure rose further, and it weakened to a tropical depression and then a remnant low late on September 3, before even reaching Bermuda.

The only effects of Fiona were some showers and gusty winds in the Northern Leeward Islands and Bermuda.



Fiona at peak intensity on September 1, despite the exposure of its circulation.



Track of Fiona, notable for coinciding almost exactly with that of Tropical Storm Colin earlier that year.

Hurricane Earl (2010)

Storm Active: August 25-September 4
On August 23, a strong tropical wave emerged off of Africa and immediately began to show signs of organization. The wave developed a low pressure center on August 24, and during that day, brought rain and gusty winds to the Cape Verde Islands. However, the system did not possess a closed circulation until August 25, and was then declared Tropical Depression Seven. Upon formation, Seven was already on the verge of tropical storm intensity and in another six hours, during the afternoon on August 25, the system became Tropical Storm Earl.

Overnight, the outflow of the system grew very organized, suggesting strengthening, but the location of the center itself was a moving target, reforming every few hours in a slightly different location relative to the convection. The center became more defined with a burst of convection during the evening of August 26, but the system did not undergo significant intensification overnight. Earl persisted westward during the day of August 27, and tropical storm watches were issued for portions of the northern Leeward islands as a result. Meanwhile, Earl began to strengthen, reaching strong tropical storm intensity by August 28. Despite an early turn north on the models, Earl continued west much longer than expected, and continued strengthening. Earl attained hurricane strength on August 29, and hurricane watches and warnings were issued for parts of the Northern Leeward Islands.

Finally, Earl turned westnorthwest later that day, but the outer bands of Earl began to sweep across the northeasternmost islands of the Caribbean bringing heavy rains and wind, with conditions getting progressively worse into the evening hours. By 8:00 pm EDT that night, tropical storm force surface winds covered the northern Leeward Islands, and hurricane force sustained winds also brushed these areas causing intense storm surge and flooding. Meanwhile, Earl continued to gain strength and rapidly became a Category 2 very late on August 29 and was on the verge of major hurricane strength by the morning of August 30. An eye appeared in Earl during the day as it strengthened rapidly, becoming a major hurricane quickly and then a Category 4 as it passed north of the U.S. Virgin Islands and Puerto Rico and caused tropical storm force winds and rain throughout the regions. Earl's pressure continued to drop, and by August 31, Earl had reached an amazing intensity of 135 mph winds and a 931 mb pressure. It still maintained a general westnorthwest motion, and the eye clouded over somewhat as Earl went through the Eye Replacement Cycle. The pressure rose as the cycle progressed, and Earl turned northwest, passing east of the Bahamas. But Earl maintained a Category 4 intensity until September 1, when it encountered some more significant shear and weakened to a Category 3 hurricane. Earl slowly turned to the north-northwest and recovered an eye, becoming more organized during the afternoon of September 1, and it restrengthened into a Category 4 hurricane. It surpassed its previous peak in intensity and reached its primary peak of 145 mph winds and a pressure of 928 mb early on September 2!

The storm continued to approach the Outer Banks of North Carolina during the day. By that evening, rainbands and tropical storm force winds swept over Cape Hattaras and the surrounding areas, as Earl turned north-northeast. However, hurricane force winds stayed offshore. As Earl approached, the eye clouded over again and Earl began steadily weakening, to a Category 3, and then a Category 2 by the time it passed by Cape Hattaras early on September 3. The weakening continued, and Earl was a tropical storm by the time it brushed passed Cape Cod overnight, bringing tropical storm force winds and rain to that area as well. By September 4, conditions were deteriorating in Nova Scotia. During the day, Earl made landfall in Nova Scotia and then Prince Edward Island as a powerful tropical storm, before entering the Gulf of St. Lawrence and finally becoming extratropical late on September 4 just off the coast of northeast Quebec. In total, Earl caused $150 million in damages and 3 fatalities over the areas it affected.



Earl nearing its peak intensity east of the Bahamas on September 1.



Track of Earl.

Hurricane Danielle (2010)

Storm Active: August 21-30
On August 19, a broad area of low pressure formed just of the coast of Africa and quickly developed deep convection due to its proximity to the Intertropical Convergence Zone. Over the next day, the system developed two centers along the trough, one on the eastern side, toward Africa, and one on the western side. The one on the western side was 1008 mb, as opposed to the eastern's 1011 mb, and the former soon gained dominance as the other dissipated. During the day of August 20, the system remained disorganized. However, during the afternoon on August 21, a closed circulation formed, and a very apparent spin appeared on satellite images, and the disturbance was classified Tropical Depression Six that day with 30 mph winds and a pressure of 1008 mb. Six strengthened as it moved westnorthwest, and a burst of convection near the center merited an upgrade to Tropical Storm Danielle during the afternoon of August 22.

Favorable conditions with warm water and minimal wind shear allowed Danielle to strengthen significantly through the night and into August 23. By the afternoon of that day, Danielle reached hurricane strength and was still rapidly intensifying. Also, contrary to previous models, Danielle still continued on a generally westnorthwestward track overnight and into the next day. During the early morning of August 24, Danielle reached Category 2 hurricane strength. However, a dry air mass embedded itself in the system during the afternoon, briefly exposing the center! This caused Danielle to weaken to a tropical storm by the evening, but already it had recovered and started to regain strength. By early on August 25, Danielle's movement slowly was shifting to the northwest, although it was still westnorthwest for much of the morning.

The system was also a hurricane again by this time and gaining intensity. Danielle maintained an intensity of 85 mph winds and a pressure of 982 mb through the day, and turned northwest during the evening. Also, an eye feature began to develop during the night, albeit an asymmetrical one, as Danielle once again became a Category 2 hurricane. The eye had been clouded over due to the Eye Replacement Cycle, but Danielle still gained intensity into August 26. Danielle redeveloped a well-formed eye during the day, and its movement to the northwest slowed as a trough interfered with its motion. Danielle continued to strengthen, becoming the first major hurricane of the 2010 season at 2 am EDT on August 27, and became a Category 4 just three hours later with an amazing intensity of 135 mph winds and a minimum central pressure of 946 mb. Later that day, Danielle achieved its peak intensity of 135 mph winds and a central pressure of 942 mb.

After that, Danielle began to be exposed to some wind shear and cooler waters, resulting in some weakening. Danielle lost its major hurricane status early on August 28, and continued its downward trend during that day as it turned to the north. During the day of August 28, Danielle, despite being over 1000 miles from the east coast, influenced the surf along the coastline, and created 3-6 foot waves and rip currents, killing one person in Florida. However, a trough moving off the east coast picked up Danielle and began to steer it to the east. That evening, Danielle's circulation broadened and became asymmetrical, marking the beginning of its extratropical transition. This transition continued into August 29, as the cyclone accelerated northeast and weakened to a Category 1 hurricane. By August 30, it was clear that Danielle was nearly extratropical and barely holding on to minimal hurricane strength, but it somehow stayed tropical through the day and turned more eastward, weakening to a tropical storm. However, by 11:00 pm EDT on August 30, Danielle had become fully extratropical and the last advisory was issued.

Danielle's remnants became embedded in a frontal boundary the next day, and it dissipated soon after as it sped off to the east. No damage and 1 indirect death occurred from Danielle.



Danielle near peak intensity over the open waters of the Atlantic.



Track of Danielle.

Tropical Depression Five (2010)

Storm Active: August 10-11
On August 8, a stationary frontal boundary off the east coast of the United States developed a low pressure system at its southern end. The force of high pressure systems to the west pushed the low south, where it encountered the typical west to east motions of the lower latitudes, and tracked over Florida on August 9, emerging over the Gulf of Mexico on August 10 as a 1010 mb low. The pressure continued to drop, and the circulation became organized enough to be declared Tropical Depression Five at its peak intensity of 35 mph winds and a pressure of 1007 mb. However, the convection associated with the system never attained definition with respect to the center, and the low weakened as it moved northwestward. The low dissipated before even reaching the Gulf coast on August 11. The broad area of low pressure associated with the dissipated Tropical Depression Five combined with a stationary front inland over the Gulf states in August 13. The system once again tracked southeastward, and approached the Gulf, intensifying as it went. As the low deepened, it became detached from the front, and by early on August 16, the system was a powerful 1010 mb low entering the Gulf with a fairly impressive clump of convection. However, no closed circulation formed, and the low made landfall once again in Louisiana without achieving tropical characteristics. The low moved north and dissipated.



Five in the Gulf of Mexico.



Track of Five.

Tropical Storm Colin (2010)

Storm Active: August 2-8
On July 29, a low pressure system formed in the southeast Atlantic and slowly drifted westward. The low became associated with a large area of showers and thunderstorms, but the system remained disorganized. A tropical wave accompanied the low as it drifted westward, but the wave disengaged from the circulation, and convection decreased. However, another tropical wave, moving off of Africa combined with the low during the day of July 31, and the systems had totally merged by August 1. However, this time convection persisted within the system and it organized. A flare up of convection that defined the system's center marked the formation of a closed circulation and the system was declared Tropical Depression Four on August 2. Tropical Depression Four's initial movement was swift, to west at 17 mph, due to the steering force of a subtropical ridge to its north. Overnight, Four became more organized, and it was upgraded to Tropical Storm Colin, with 40 mph winds and a pressure of 1006 mb, during the morning of August 3.

Colin accelerated to the west at an even greater speed, reaching a velocity of nearly 25 mph during the day on August 3. Then, the low pressure associated with the system became open, and it degenerated into a remnant low. However, cloud cover persisted within the system, and a new surface low pressure center became evident late on August 4. The system continued to organize, and was redesignated Tropical Storm Colin on August 5. Colin moved northwest and slowed in movement as it encountered a high pressure system, and began to turn east, as with most cyclones in the region. It briefly attained an intensity of 60 mph winds and a pressure of 1005 mb before the circulation became widely separated from the cloud cover again and Colin weakened to a weak tropical storm (45 mph winds) by August 6. Colin became nearly stationary on August 7, and weakened further, barely a tropical storm by the time it resumed movement to the northnortheast later that day. As Colin approached Bermuda, it weakened into a tropical depression, and brought needed rain to the island. Soon after passing west of Bermuda on August 8, Colin lost its circulation and dissipated.



Tropical Storm Colin shortly after reforming on August 5. The circulation, although more organized than before, is clearly exposed.



Track of Colin.

Tropical Storm Bonnie (2010)

Storm Active: July 22-24
On July 14, a tropical wave emerged off of Africa and moved westward. By July 17, the wave became associated with a broad upper-level low. However, very little storm activity accompanied the system at that time, as it moved westnorthwestward. On July 18, cloud cover increased in the system, but the circulation remained in the upper levels, prohibiting development. Over the next few days, the surface pressure began to drop and heavy rain from the system caused widespread flooding in Puerto Rico and Hispaniola. On July 22, a surface circulation appeared, and the system was upgraded to Tropical Depression Three, with 35 mph sustained winds and a pressure of 1008 mb just north of eastern Cuba. However, shower activity was primarily displaced to the north and east of the center due to wind shear.

Despite wind shear, a small intensification of the system during the evening of July 22 allowed the system to develop into Tropical Storm Bonnie, with 40 mph sustained winds. Due to an upper level ridge to the system's north, Bonnie accelerated to the westnorthwest during the morning of July 23, reaching a forward speed of 19 mph by 8:00 am EDT that morning. Soon after, the system slammed into Florida with 40 mph winds. Bonnie lost most of its convection before entering the Gulf, and was downgraded to a tropical depression. Despite a redevelopment of convection overnight, the surface pressures continued to rise and the system's center was stripped away by shear, leaving a exposed circulation. Bonnie continued struggling northwestward through the Gulf of Mexico during the morning of July 24, but it ultimately degenerated to a remnant low later that day. The low made landfall in Louisiana on July 25 as it dissipated. Damage was minimal, and one death was recorded in association with this system.



Bonnie after landfall in Florida.



Track of Bonnie.

Tropical Depression Two (2010)

Storm Active: July 7-8
The tropical wave that eventually became Two formed over northeast South America within the ITCZ (Intertropical Convergence Zone). It slowly moved northwestward and as it moved into the western Caribbean, scattered thunderstorm activity began to be associated with it. However, this activity remained disorganized, and the system tracked over the Yucatan Peninsula on July 6. Although the system lost a lot of cloud cover over land, a low pressure center actually developed during this time, allowing the system to be more organized as it emerged into the Gulf of Mexico on July 7. However, the low pressure became elongated over the next few hours, and did not assume the perfect circular shape reminiscent of a healthy circulation. Nevertheless, convection continued to organize, and late that night the system was declared Tropical Depression Two with 35 mph winds and a minimum pressure of 1005 mb.

As had been the trend for a few days, the system lost much of its convection overnight, as it moved towards the Texas-Mexico border at 12 mph, but the circulation remained intact, and even strengthened a little. However, the system did not have enough time to reach tropical storm strength and made landfall in northern Mexico at 11:15 a.m. EST on July 8. The system quickly dissipated over land that evening. Overall, the main effect of Two was flooding, as it hit in an area which had already suffered from Hurricane Alex a few days eariler.



Two at landfall.



Track of Two.

Hurricane Alex (2010)

Storm Active: June 25-July 1
On June 12, a strong tropical wave moved off of Africa and persisted westward. For the next week, it was held within the Intertropical Convergence Zone, and showed no signs of development. On June 20, the tropical wave moved into the Caribbean, and moved slightly northward on its westerly track. The wave tapped into the large amount of moisture in the Caribbean and developed a broad area of convection with little organization. The wave continued through the Caribbean, encountering increasingly favorable conditions as it went, and slowly organized. On June 24, a low became associated with the tropical wave, but most convection was still to the east of the circulation. However, the convection soon concentrated at the center of the system, and soon developed an apparent circulation. On June 25, an aircraft reconnaissance mission confirmed the existence of a closed circulation and the system was declared Tropical Depression One off the coast of Honduras.

Tropical Depression One continued to gain organization, and was declared Tropical Storm Alex the following morning as it moved westnorthwest at 10 mph towards the Yucatan Peninsula. During the day of June 26, Alex assumed a more westerly track, and gained intensity as its thunderstorm activity concentrated toward the center. As a result, Alex reached an intensity of 65 mph winds and a central pressure of 996 mb, before making landfall in central Belize at approximately 9:00 pm EST June 26. Alex remained over land for the next day, weakening as it went, and by the morning of June 27, Alex was downgraded to a tropical depression. Alex maintained its impressive circulation, but cloud cover depleted, as the system did not receive new water to fuel itself as it moved westnorthwestward at 12 mph. Alex slowed down and turned more to the northwest as it emerged into the Gulf during the evening of June 27, allowing for more time over the favorable Gulf, and more strengthening. As Alex reemerged over water, deep convection immediately started to appear around the center, and the system was once again upgraded to a tropical storm overnight, as it moved slowly northwest. Mild shear affected the system, but it strengthened nevertheless during the day of June 28. However, the shear lessened on June 29, and the system strengthened to a hurricane overnight.

Alex began to turned westward again, as its outer bands swept across the coast of northern Mexico and southern Texas. Alex continued west, and strengthened further, reaching its peak intensity of a category 2 hurricane with 105 mph winds and a minimum pressure of 947 mb just before landfall in northern Mexico at 10:00 pm EST June 30. Alex then began weakening, becoming a category 1 hurricane by very early the next morning, and a tropical storm a few hours later. All warnings and watches were quickly discontinued as the convection associated with Alex continued to weaken. By the late evening of July 1, Alex's circulation and vorticity were gone, and the system had dissipated. However, Alex caused $1.21 billion in damages, and 32 deaths were associated with the system. As a result, Alex was already much more costly and much more deadly than the entire 2009 Atlantic hurricane season! Also, Alex was a fairly rare event climatologically, being the first June hurricane since 1995, and the most powerful in central pressure since 1957.



Image of Alex near peak intensity just before landfall in Mexico.



Track of Alex.

Professor Quibb's Picks-2010

My personal picks for the 2010 Atlantic hurricane season:

18 cyclones attaining tropical depression status
17 cyclones attaining tropical storm status
7 cyclones attaining hurricane status
4 cyclones attaining major hurricane status

Hurricane Names List-2010

For the Atlantic Basin in 2010, the names list is as follows

Alex (used)
Bonnie (used)
Colin (used)
Danielle (used)
Earl (used)
Fiona (used)
Gaston (used)
Hermine (used)
Igor (used)
Julia (used)
Karl (used)
Lisa (used)
Matthew (used)
Nicole (used)
Otto (used)
Paula (used)
Richard (used)
Shary (used)
Tomas (used)
Virginie
Walter

This list is the same as the one used in 2004, except for Colin, Fiona, Igor, and Julia, which replaced the four retired hurricanes of 2004: Charley, Frances, Ivan and Jeanne.

Polytopes: Part IV

This is the final part of a four part post. For the first part, see here. For the second part, see here. For the third part, see here.

All the regular polytopes up through the fourth dimension have been discussed, and we couldn't directly visualize these elaborate structures, but we could understand their construction, and their polyhedral components. However, when one goes to the fifth dimension, all direct understanding is out of reach. And what of the sixth dimension? And the seventh? And the hundredth? How can we possibly deal with the polytopes in these dimensions? However, rather than increasing in complexity, regular polytopes become much simpler in higher dimensions, which allows us to generalize to n-dimensions in many aspects.

Starting with five dimensional space, we must consider a further extension of the Schlafli system. Since the structures in four dimensions were denoted by {p,q,r}, we now have five-dimensional polytopes {p,q,r,s}. Based on the sixteen regular polychora, ({3,3,3}, {4,3,3}, {3,4,3}, {3,3,4}, {5,3,3}, {3,3,5}, {5/2,3,3}, {3,3,5/2}, {5/2,5,3}, {5/2,3,5}, {5,5/2,5}, {3,5,5/2}, {5,3,5/2}, {5/2,5,5/2}, {5,5/2,3}, and {3,5/2,5}) one arrives at a staggering 34 possible forms:

{3,3,3,3}, {3,3,3,4}, {3,3,4,3}, {3,4,3,3}, {4,3,3,3}, {3,3,3,5}, {5,3,3,3}, {4,3,3,4}, {4,3,3,5}, {5,3,3,4}, {5,3,3,5}, {3,3,3,5/2}, {5/2,3,3,3}, {4,3,3,5/2}, {5/2,3,3,4}, {5,3,3,5/2}, {5,5/2,3,3}, {5/2,3,3,5}, {5,5/2,5,3}, {3,5/2,5,3}, {3,5,5/2,5}, {5,3,5/2,5}, {3,3,5,5/2}, {5/2,5,3,3}, {5,5/2,3,5}, {3,5,5/2,3}, {5/2,5,5.2,5}, {5/2,5,5/2,3}, {3,5/2,5,5/2}, {5,5/2,5,5/2}, {5/2,3,3,5/2}, {5/2,3,5,5/2}, {5/2,5,3,5/2}, {3,3,5/2,5}

11 of these involve only convex polychora, while the remaining 23 involve star polychora. Despite the vast range of possible forms, very few actually create polytopes in the fifth dimension. These are called 5-polytopes, or polytera (singular: polyteron). The curvature of these polytera is defined by determining whether the four dimensional solid angle around each vertex adds up to more than, less than, or exactly equal to, the four dimensional sphere. To find the curvature equation in higher dimensions, it becomes useful to use the general form of the equation in n dimensions. The function to find the formula is known as the ∆ (delta) equation. The curvature formula for a polytope {p,q,r...,y,z} (with any number of letters in between r and y) is expressed ∆{p,q,r...,y,z}, and depends whether the resulting formula is greater than, less than, or equal to zero. The ∆ equation is defined recursively, or that each formula depends on the previous one counting up in dimensions. Assuming the trivial cases in one and two dimensions as follows (with {} implying the straight line as the universal polytope in one dimension):

∆{}=1 (this formula never changes in value, because all polytopes in one dimension are lines and are all basically identical)
and
∆{p}=(sin(π/p))^2 (this formula is always positive, corresponding to the fact that polygons always have positive curvature and are finite)

one can find the formula for any number of dimensions greater than two using:

∆{p,q,r...,y,z}=∆{q,r,...,y,z}-∆{r,...,y,z}*(cos(π/p))^2

For example, to find ∆{p,q}:

∆{p,q}=∆{q}-∆{}*((cos(π/p))^2)=
(sin(π/q))^2-1*((cos(π/p))^2)=
(sin(π/q))^2-(cos(π/p))^2

By setting this greater than zero (solutions would then be finite polyhedra)

(sin(π/q))^2-(cos(π/p))^2>0
(sin(π/q))^2>(cos(π/p))^2
sin(π/q)>cos(π/p)

which, by a property of trigonometry, (for p,q>2, which, conveniently is what is required for true polyhedra) equals

sin(π/q)>sin(π/2-π/p)
π/q>π/2-π/p
π/p+π/q>π/2
1/p+1/q>1/2

The final formula seems very familiar, as it is the curvature formula from the second part of this post for polyhedra, that we have successfully derived using the ∆ formula! Using the same equation for ∆{p,q,r,s} (I won't show all the work this time), we obtain the curvature formula for polytera. For a finite polyteron,

((cos(π/q))^2)/((sin(π/p))^2)+((cos(π/r))^2)/((sin(π/s))^2)<1

12 of the 34 total forms satisfy this: {3,3,3,3}, {3,3,3,4}, {4,3,3,3}, {3,3,3,5/2}, {5/2,3,3,3}, {4,3,3,5/2}, {5/2,3,3,4}, {5,5/2,3,3}, {3,3,5,5/2}, {5/2,5,5/2,3}, {3,5/2,5,5/2}, and {5/2,3,3,5/2}. However, all nine of these that are star polytera can be calculated to have infinite density, meaning that there are infinite planes in the polyteron. However, this is impossible in regular finite polytera, and therefore all but the first three can be eliminated. We will return to more general forms of {3,3,3,3}, {4,3,3,3} and {3,3,3,4} later.

There can also be tilings of Euclidean four dimensional space and these are the only ones that can be understood in four dimensions. The simplest example, {4,3,3,4}, also known as the tesseractic honeycomb, has four tesseracts (8-cells) at each face, and a three dimensional projection is shown below.



With these figures, it is difficult to see any recognizable structure, but the 8-cells in this picture can vaguely be seen.

Similarly, {3,4,3,3} has three 24-cells at each face, and {3,3,4,3} has three 16-cells at each. No star polytera exist that are tilings of the Euclidean four dimensional plane, although {5,3,3,5/2}, {5/2,3,3,5}, {3,5/2,5,3} {3,5,5/2,3}, {5/2,5,5/2,5}, {5,5/2,5,2/2}, {5/2,3,5,5/2} and {5/2,5,3,5/2} all satisfy

((cos(π/q))^2)/((sin(π/p))^2)+((cos(π/r))^2)/((sin(π/s))^2)=1

However, all nine possible four dimensional hyperbolic tilings exist, namely: {3,3,3,5}, {5,3,3,3}, {4,3,3,5}, {5,3,3,4}, {5,3,3,5}, {3,5,5/2,5}, {5,5/2,5,3}, {3,3,5,5/2}, and {5/2,5,3,3} and all these satisfy

((cos(π/q))^2)/((sin(π/p))^2)+((cos(π/r))^2)/((sin(π/s))^2)>1

Finally, returning to the three finite regular polytera {3,3,3,3}, {3,3,3,4} and {4,3,3,3}, one can see that only four forms are possible in six dimensions: {3,3,3,3,3}, {3,3,3,3,4}, {4,3,3,3,3} and {4,3,3,3,4}. The last of these is a tiling (as we will soon see) and the only seven dimensional regular figures are {3,3,3,3,3,3}, {3,3,3,3,3,4}, {4,3,3,3,3,3} and {4,3,3,3,3,4}. The last of this is also a tiling, and the pattern continues. Therefore, for finite regular polytopes existing in n dimensional Euclidean space (n-1 dimensional elliptic space) there are only three forms:

{3^(n-1)}
The general polytope in n dimensions with n-1 3's in its symbol is known as the n-simplex. It has n+1 vertices and the rest of its elements, known as i-faces, come in numbers discussed below. The n-simplex is always a regular finite polytope in any number of dimensions.

Construction in n dimensions: Start with a point. This is the 0-simplex. Choose another distinct point and connect them. The result is a line segment, which is the 1-simplex. Choose a point outside of this line that is equidistant from the two existing points and connect them. The result is the regular triangle {3}, which is the 2-simplex. Choose another point outside this plane, that is equidistant from all three points and connect each pair of points with an edge. The result is the tetrahedron {3,3} which is the 3-simplex. Continue this procedure for any number of dimensions.

The number of i-faces (0-face=vertex, 1-face=edge, 2-face=face, 3-face=cell, etc.) in an n-simplex is based on the binomial theorem and Pascal's triangle. For an n-simplex, the number of i-faces is

(n+1)!/((i+1)!(n-i)!)

where ! is the factorial function (n!=1*2*3*4...*n).

For example, the number of faces on a 4-simplex is 5!/((3!)(4-2)!)=10, and the number of 5-faces in a 9-simplex is 10!/((6!)(9-5)!)=210.

The n-simplex is the simplest polytope that needs n dimensions to define, and is the general form of the sequence: point, line segment, triangle, tetrahedron, pentachoron... etc. In addition, n-simplices are most often represented by symmetric graphs that map out the vertices and show the connections between them. However, the drawback of this representation is that only vertices and edges can be mapped, and there is no easy way to see higher i-faces. As a result, symmetric graphs give few implications to the actual structure of the polytope. The symmetric graph of the 5-simplex {3,3,3,3} is shown below.



The symmetric graph of the 5-simplex (or hexateron). The six vertices and 15 edges are visible, as is the fact that every pair of points is connected, but this view lacks any higher features. Also, for an n-simplex, the symmetric graph is always based on the regular polygon {n+1}.

{4,3^(n-2)}
The general polytope in n dimensions with a 4 followed by n-2 3's in its Schlafli symbol is known as the n-cube. It is always a finite regular polytope in any number of dimensions. It has 2^n vertices and the number of i-faces again depends on the binomial theorem but with an extra term in front of it. In general, to find the number of i-faces of a n-cube, the formula is

(2^(n-i))((n!)/((i!)(n-i)!))

For example, the number of edges (1-faces) on a 3-cube is (2^(3-1))((3!)/((1!)(3-1)!))=12, and the number of 7-faces on an 18-cube is (2^(18-7))((18!)/((7!)(18-7)!))=65175552.

Construction in n dimensions: To construct a regular n-cube, begin with a point. This is the 0-cube. Choose another distinct point and connect them. The result is a line segment, which is the 1-cube. Define another line segment as the shifting of the original one which is parallel to the original and so the distance between the lines is the same as the length of each line segment. Connect the corresponding vertices on the two line segments, and the result is the square {4}, which is the 2-cube. Take this square, and shift it out of the existing plane up or down the distance between any two points, to obtain two parallel squares. Connect each original point to its corresponding shifted point. The result is the 3-cube {4,3}. Continue for any number of dimensions to obtain any n-cube. To demonstrate this procedure, the construction for the 4-cube {4,3,3}, also called the 8-cell, is shown below.



n-cubes, just like n-simplices, can be expressed with symmetric graphs. For example, the 6-cube's symmetric graph is shown below.



The n-cube is also called the measure polytope in n dimensions, and is the general form of the sequence point, line segment, square, hexahedron, octachoron... etc.

{3^(n-2),4}
The general polytope in n dimensions with n-2 3's followed by a final 4 in its Schlafli symbol is known as the n-orthoplex, or the cross polytope. It is the general dual of the n-cube, and has 2n vertices. A general form for the number of i-faces is once again based on the binomial theorem. In general, the number of i-faces in an n-orthoplex is

(2^(i+1))((n!)/((i+1)!(n-(i+1))!))

For example, the number of cells in a 5-orthoplex is (2^(4))((5!)/((4)!(5-(5))!))=80, and the number of 6-faces in a 10-orthoplex is (2^(7))((10!)/((7)!(10-(7))!))=15360.

Construction in n dimensions: Start with a point. This is the 0-orthoplex. Choose another distinct point and connect the two. The result is the 1-orthoplex, or a line segment. Then choose two points equidistant from the existing two and connect them to the existing points in such a way that a square is formed (eliminate the line between the original two points). The square {4} is the 2-orthoplex. Then, choose a point equidistant from all four of the existing points that is the same distance from each point as each point is from another adjacent to it. Also choose this point's reflection through the square. Connect all points, except those directly opposite from each other. The result is the octahedron {3,4} which is the 3-orthoplex. This is a more complex construction process, but it is easy to see with a little thought that it produces the orthoplexes. This process can be continued for any number of dimensions.

As with the first two general polytopes, the n-orthoplex can be represented with a symmetric diagram. For example, the layout of the 7-orthoplex's 14 vertices into the regular polygon {14} looks like this:



This symmetric graph looks similar to that of a simplex, but with one exception. The vertices in a simplex are all connected in every possible way, but in an orthoplex, points are connected to every other point except the one directly opposite it. In conclusion, the n-orthoplex is a convex regular finite polytope in any number of dimensions, and is the general term of the sequence point, line segment, square, octahedron, 16-cell, etc.

{4,3^(n-2),4}
The fourth and final regular polytope in n-dimensions is represented by a 4, followed by n-2 3's, and then another 4. It is an infinite cubic honeycomb in Euclidean geometry, which is why {4,3^(n-2),4} needs only n dimensions to be visualized. If it were an elliptic or hyperbolic polytope, it would need n+1 dimensions, and the symbol would be {4,3^(n-3),4}. The elements of the n-cubic honeycomb are simply n-cubes.

After five dimensions, the above four are the only regular polytopes for each dimension, but there are many other polytopes with regular elements, and these are known as uniform polytopes. One subset of uniform polytopes is the regular ones discussed above, and another is the quasi-regular polytopes, which are based on two types of i-faces. There are limited possibilities for these, and these come in a few different families, each represented by a special Schlafli symbol {3^{a,b,c}}. Of the four families, only one provides an infinite number of quasi-regular polytopes. Note that the {a,b,c} doesn't tell the number of 3's, as before, but has a different meaning, which is not discussed here.

{3^{1,b,1}}
The first family of quasi-regular polytopes is represented {3^{1,b,1}}, where the variable b is used just to match up with the {a,b,c} above. These polytopes are known as n-demicubes. In n dimensions, the n-demicube is {3^{1,n-3,1}}. They are constructed by connecting alternating vertices of an n-cube with edges. To demonstrate this process, the 3-cube and 3-demicube are shown below.



This image shows a transparent cube (3-cube) with its two possible demicubes. One connects alternate vertices, and the second simply connects all those not covered by the first. The two demicubes of a cube are always identical, and in this case, are two tetrahedra. Therefore, the tetrahedron, as well as being {3,3}, is also {3^{1,0,1}}. The n-demicube is quasi-regular beginning with the fifth dimension, as the 4-demicube is the 16-cell. Each n-demicube is made up of (n-1)-simplices and (n-1)-demicubes, and therefore is also defined recursively, with each demicube depending on the one before it. Each n-demicube has exactly 2n (n-1)-demicubes and 2^(n-1) (n-1)-simplices as (n-1)-faces. These are the highest faces before the polytope itself, and are sometimes called facets.

Since the 3-demicube is a tetrahedron, and the 3-simplex is also a tetrahedron, the 4-demicube is made of 8 tetrahedra and 8 tetrahedra, and the result is 16 tetrahedral cells, which is a 16-cell. However, the 5-demicube is made out of 10 4-demicubes, which are 16-cells, and 16 4-simplices, which are 5-cells, and the 5-demicube is therefore the first quasi-regular demicube. Continuing this pattern, one finds that there are infinite demicubes, all of which are finite and quasi-regular. As before, n-demicubes are also represented with symmetric graphs, and the 7-demicube is shown below in symmetric graph form.



The 7-demicube has only half the vertices of the 7-cube, but over 200 more edges. Also, red dots represent single vertices, while orange represent two overlapping and yellow four.

{3^{1,b,2}}
The second family of quasi-regular polytopes is the {3^{1,b,2}} family. The {3^{1,b,2}} polytope only is distinct from aforementioned polytopes when b takes the values 2, 3, 4 and 5, corresponding to polytopes in 6, 7, 8, and 8 dimensions, as we will see shortly. When b takes the value 0, the {3^{1,0,2}} polytope exists in four dimensions, and is simply {3,3,3}, or the 4-simplex. From there, the system is defined recursively, with each {3^{1,b,2}} in n dimensions (with one exception, see below) having {3^{1,b-1,2}'s and (n-1)-demicubes as facets. The polytope {3^{1,1,2}} has 4-simplices and 4-demicubes as facets, and this has already been discussed above as being the 5-demicube. Therefore,

5-demicube={3^{1,2,1}={3^{1,1,2}}

However, the {3^{1,2,2}} polytope has {3^{1,1,2}}'s and 5-demicubes as facets, which are both equivalent, and the {3^{1,2,2}} polytope is therefore a new polytope, made of 54 5-demicube facets. The next polytope {3^{1,3,2}} has {3^{1,2,2}}'s and 6-demicubes as facets, and the pattern continues. However, when one gets to {3^{1,5,2}} polytope, composed of {3^{1,4,2}}'s (in eight Euclidean dimensions, seven elliptic) and 8-demicubes, one finds not an elliptic polytope, but a new Euclidean tiling! Therefore, this infinite polytope also exists in eight dimensions. However, having a Euclidean tiling in the polytope family ends it, for if the {3^{1,6,2}} polytope existed, it would have to have {3^{1,5,2}}'s as facets, and these are infinite, which is not allowed in polytopes. Therefore, only four new polytopes arise from this family. The symmetric graph of {3^{1,3,2}} is pictured below.



In this image, blue vertices are single and red have a multiplicity of 2 (having 2,4,6 or some even number of vertices coinciding).

{3^{2,b,1}}
The third of four families of quasi-regular polytopes is the {3^{2,b,1}} family. Again, only four new polytopes are generated by this family and these are when b=2, 3, 4, or 5. The facets of such a polytope in n dimensions (again, with one exception) are {3^{2,b-1,1}}'s and (n-1)-simplices. The {3^{2,0,1}} polytope is the 5-cell again and the {3^{2,1,1}} has 32 4-simplices, making it the 5-orthoplex. The first new polytope is {3^{2,2,1}} which has 5-orthoplexes and 5-simplices as facets. The pattern continues again until {3^{2,5,1}} which is another Euclidean tiling, made of an infinite number of 8-simplices and {3^{2,4,1}}'s. As before, the pattern must end there, since there cannot be a polytope with infinite facets, although a polytope can have an infinite number of facets. The {3^{2,4,1}} polytope is shown below.



This polytope exists in eight dimensional Euclidean space, and only its 2162 vertices are presented in this graph. Although it may seem that presenting only vertices doesn't present much of the polytope, the addition of edges would over clutter the image, as the {3^{2,4,1}} has over 69000!

{3^{a,2,1}}
The final quasi-regular polytope family is the {3^{a,2,1}} family and it is the only one of the four in which the first number a of the symbol varies and not the second, b. This family provides only 3 new polytopes in 5 dimensions and up, but yields a few interesting cases in lower dimensions as well. In n dimensions, the facets are simply (n-1)-simplices and (n-1)-orthoplexes. The sequence begins with the {3^{-1,2,1}} polytope, which exists in three dimensions and has triangles and squares as faces. The resulting figure is a triangular prism, which is simply two parallel triangles connected by three squares. The next polytope has 3-simplexes and 3-cubes as cells and is represented {3^{0,2,1}}. This polytope is equivalent to the rectified 5-cell. The next two, {3^{1,2,1}} and {3^{2,2,1}}, have been covered already. The first of these is the 5-demicube, and the second is discussed above and is a member of the {3^{2,b,1}} family.

The first distinct polytope is the {3^{3,2,1}} polytope which is made up of {3^{2,2,1}}'s and 6-orthoplexes. The pattern continues until, as before, the {3^{5,2,1}} polytope is a Euclidean eight dimensional tessellation with infinite {3^{4,2,1}}'s and 8-orthoplexes as facets. As before, {3^{6,2,1}} cannot exist as a result.



The symmetric graph of the eight-dimensional (seven-dimensional elliptic) {3^{4,2,1}} polytope.

No other major families of quasi-regular polytopes exist, because all of the possibilities of recursive dependance, i.e. being based on the previous polytope and the simplex, or the orthoplex and simplex, etc. None involve square faces or any n-cubes at all, except of course, the n-cubes themselves, and are all based on triangles. However, there are many other uniform polytopes that may be obtained by operations on the three regular polytopes in n-dimensions. It has been discussed numerous times in the previous posts that truncation, or the slicing of vertices, and cantellation, or the slicing of edges. Also, in the previous part of this post, the idea of the slicing of cells in four dimensions and up, called runcination, was discussed. These operations can be extended, however, to the slicing of any i-face, and there are more possibilities in every dimension. For example, in five dimensions, the sterication operator, or slicing of 4-faces with respect to the fifth dimension, can be added to the four other operators, and any combination of the five can be applied. By the sixth dimension, there are hundreds of possibilities with the addition of the pentellation operator, and soon there are way too many to keep track of. Names of the polytopes also become unwieldy, as the combination of truncation, cantellation, sterication, and pentellation on a 6-simplex is named the pentistericantitruncated 6-simplex. Due to their diverse natrue, the uniform polytopes in seven dimensions and up have not been properly classified.

With any number of dimensions, and an infinity of possibilities in each, the world of polytopes is a limitless, beautiful branch of mathematics that defines much that we see in nature and what we find in abstraction.

Sources: Regular Polytopes by H.S.M. Coxeter, and various wikipedia titles: List of Regular Polytopes, k21 familiy, Uniform polyteron, Uniform polypeton, Truncation (geometry), etc.