Intensive and Extensive Quantities: How to make more useful Maps and Tables about COVID-19


We are flooded with information about COVID-19.

Unfortunately most maps and tables and diagrams work with absolute numbers. With these China and the United States almost automatically are positioned in some of the first spots.

We should think which quantities are meaningful.

Let’s take a look at Thermodynamics and Physical Chemistry. There it is quite common to use intensive and extensive quantities.

An intensive quantity describes primarily the properties of a substance, while an extensive quantity describes the amount.

Some examples:

Typical intensive quantities are temperature, density, pressure, concentration, energy density…

Typcial extensive quantities are mass, weight, volume, number fo molecules/atoms, electric charge, energy,…

Now a lot of quantities, that are themselves extensive allow deriving closely related intensive quantities. For example electrical charge is an extensive quantity. That means, whenever we double the system, the charge is exactly doulbed. For a condensator as an energy storage or as part of an electric circuit, this is useful. And we see electrical engineers working with charges and capacities.

But from the viewpoint of physical chemistry, the density of charge might be more interesting. That can be the charge per volume, per number of molecules (molarity) or per mass. A system behaves locally differently, if the density of charge is differently.

Very often we encounter that Enthalpy, which is the inner thermal energy and Entropy are used. Please check Wikipedia or books about physical chemistry for an explanation….

Both can be viewed as total quantities of a system. So they double with the size of the system. That is interesting, because the first law of thermodynamics and the second law of thermodynamics are usually formulated like that. We just consider a closed system, which can be the whole universe and there it is valid. At least we think so. At least the second law of thermodynamics has been adjusted in some way that it still works in conjuncion with black holes.

Now we can derive intensive quantities from entropy and enthalpy. Usually this is done in viewing them „per mol. That just sets them in relation to the number of molecules, just with slightly more conveniant numbers.

Often intenive quantities are way more interesting. Doubling the system should ideally not change anything, if we neglect that the system might behave differently at its borders or that it might be so small that the number of molecules introduces observable granularity. Having too large system can be bad too, because they put together too many unrelated things in one measurement.

How does this apply to the current coronavirus pandemic?

We often see charts and tables and maps that show absolute numbers. How many people got infected, how many died, how many recovered? Some of these maps are dark red for countries with many inhabitants. They use extensive quantities. Which is actually quite absured.

But we should want to use intensive quantities. We should count the cases per million inhabitants. Maybe the number of cases pare area, but this is already less interesting. So the strong advice, always display number of cases divided by the number of inhabitants (and multiplied by 1’000’000 or 10’000 for nicer numbers).

Another aspect: small countries like Switherland or Iceland can be considered as one unit, at least for a map of Europe of a world map. The question is if relatively large countries with relatively few inhabitants like Sweden or Norway should be subdivided into regions. The map could allow to regcognize differences between a denser populated south and a thinner populated north. For countries with many inhabitants and a large area like India, the United States, China, Brazil this should be done. For countries with small area and a lot of inhabitants like Singapore, Hong Kong etc. there is no point in subdividing into areas.

We should also be aware that data is very inaccurate, I would say to a factor of ten. So it is hard to compare numbers between countries, because they measure differently.

By now actually data should be sufficient to actually guess more accurately thant the official numbers of the country. By guessing I mean a team of scientists applying data and software and knowledge, not guessing out of nothing.

But now matter how accurate the data are, they already make more sense as intensive quantities and using extensive quantities is just a widespread absurdity. And of course it should always be mentioned what the numbers really measure.

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Kirnitzschtalbahn – a rural tram line

(Terje Melheim)

Tramways are usually associated with large towns, built up areas, big buildings and much traffic. Can you imagine a tram line through a fantastic rural landscape? An example of such a tram line is Kirnitzschtalbahn in Eastern Germany. That Kirnitzschtalbahn has survived till today is probably due to the fact that it was situated in former East Germany. East Germany had another economic structure than countries in Western Europe, and a tram line was more needed, and so it was allowed to remain.

The gate to the valley of Kirnitzsch is the small tourist town Bad Schandau on the river Elbe. From the town of Dresden you will reach Bad Schandau by local train (S-Bahn), but as the town itself lies on the opposite bank from the station, you have to cross the river by a ferry. Before 1963 you could reach the terminus of Kirnitzschtalbahn after a short walk. In two steps the line was shortened, and in 1969 the new terminus was established in the town park.  The position of a tram line was not too safe even in socialist German Democratic Republic. The reason for withdrawing the Kirnitzschtal from a central position in the town was that the street was needed for the motor traffic. Several times during the last century there were strong voices which wanted to replace the trams by buses. For shorter periods the tram line was closed to all traffic but it was finally decided that the the trams should remain, and necessary renewals of the track and the overhead wires were introduced. The Kirnitzschtalbahn is an essential factor for tourism in this special landscape called Saxonian Switzerland.

The standard Gotha tram of most East German towns were gradually replaced by more modern trams. The Kirnitzschtalbahn took over many Gotha trams from various towns with metre gauge tram lines and supplied  the Gotha trams with the characteristic dark yellow colour. Today the Gotha trams of the Kirnitzschtal give the tram line a nostalgic look. At both terminus trailers are strategically parked, so that by demand trailers can be coupled to the trams. Only occasionally an original tram from the Kirnitzschtalbahn is used.  I am so lucky that I have a photo of tram no. 5 (MAN) which was originally delivered to the Kirnitzschtalbahn in 1928. This photo was taken in the 1970ies when no. 5 was in regular use before the Gotha trams dominated the line.

At the terminus Stadtpark. Motor no. 4 was transferred from Lockwitztalbahn in 1977.

A tram is leaving Bad Schandau

Motor and trailer originally delivered to Kirnitzschtalbahn in 1928.

When I visited Kirnitzschtalbahn in 1992 I heard from a tram driver/conductor that the tram line would be extended through Bad Schandau to its original terminus. In 1992 disused tram tracks remained in the town. In 2001 when I again visited the tram line, no such extension had taken place. The tracks in the streets of Bad Schandau had been removed. A ribbon of black asphalt in the nostalgic cobble stoned streets showed where once the trams of the Kirnitzschtalbahn transported its passengers through the town.

At the terminus Town park (Stadtpark) an interesting plaque of iron has been put up, saying that at this site was the edge of the largest extension of the Scandinavian ice shelf. This fact explains the special landscape you find in this area. The melting water from the ice cap has effectively eroded this sand stone area. Deep valleys have been excavated in marked contrast to the rocks which have remained. The landscape is called Saxonian Switzerland. The metre gauge tram line follows the valley Kirnitzschtal, surrounded by steep rocks, which make the ride on this tram line particularly impressive. The tram line, which is single track throughout, runs along the right  hand side (from Bad Schandau) of the road through the Kirnitzschtal. The road is set with cobble stones and gives the right nostalgic look. Between the rails of the tram line the cobble stones have been replaced by plates of concrete.

On the single track a tram heading for Lichtenhainer Wasserfall is waiting for the tram towards Bad Schandau.

The tram track lies at the edge of the road in this fantastic landscape.

From the Kirnitzschtalbahn you have access to the Schrammstein area in the Saxonian Switzerland.

On its way up through the valley the tram passes the depot on the roof of which solar cells have been installed. This device delivers as much as one third of the current needed to operate the tram line. The various stops along the line are starting points for hiking tours into the Schrammstein area with its incredible rock formations. After 8 km the upper terminus is reached – Lichtenhainer Wasserfall. At this point there is a restaurant which has got tables in the open air on the other side of the tram track. In 1992 I observed a waiter carrying a tray full of drinks and food and was on his way to the guests on the other side of the track. A tram was approaching; and the driver slowed down and let the waiter pass. I found that such a nice gesture, and that is characteristic of the Kirnitzschtalbahn which leaves you with so many favourable impressions.


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How to continue with the rail line from Rovaniemi to Kirkenes?


As could be observed, the project of a railroad from Rovaniemi to Kirkenes has been put on hold, because of not being economically viable with the currently predicted amounts of freight volume. Passenger service, which would be expected to happen on this line, would be so little that it cannot contribute a large share, so the justification of the product has to come from freight traffic.

Just to address a common misunderstanding: It is not possible and not likely and not the idea of the project, to replace a large share of traffic going from China or other parts of China to European countries like Germany, France, Italy, that are far away from Kirkenes. A reasonable goal would be to cover a large part of trade between Finland, northern Norway, northern Sweden and a bit from the Baltic countries and Russian areas near the northern Finnish border. So the countries having the rail line would mostly profit from it, in the case of Finland of course including the more densly populated south. Slogans that predict that Kirkenes will become the „new Rotterdam of the north“ are exaggerated or have to be read like „Kirkenes will be for Northern Europe what Rotterdam is for the rest of Europe“.

Also there seems to be resistance from the Sámi people. I have a hard time accepting their style of arguing that this rail line would be a genocide, because this word may well be used for circumstances that destroy a culture and a people without physically killing it, but not for building a single rail line in an area that is already accessible by roads. The argument is, that the line would cut the grassing areas of reindeer and thus destroy the traditional culture of reindeer herding. Why should a rail line that is running near the Russian border have such an effect? The area already has a road network that cuts the area in parts or endangers the animals crossing them and the border between Norway or Finland and Russia, which is not very open. It is possible to build short tunnel sections in a railroad line to allow animals to cross, even without mountains that would require tunnels. Bridges would also be possible. This is something that is sometimes done for new roads and railroad in some countries, for example in Switzerland. It would of course be necessary to provide such crossings for this rail line as well. Why not for the roads? It is assumed that road vehicles can stop on sight when animals are on the road, which is not possible for trains, so a fence and safe crossings would be needed. I do not know if this was offered as part of the plan. Unfortunately the Sami representatives do not seem to show any willingness to compromises.

Freight traffic that could be performed by the railroad is still taking place, partially by trucks on the roads, partially on other railroad lines. An expanded port of Kirkenes could have a high percentage of its land transport done by rail, because rail transport is efficient for long distances with a single point like a port as destination. And the sea routes could be shorter as well when crossing the northern Atlantic or circumnavigating northern Siberia and the Bering Strait to East Asia and the North American Pacific coast.

Many factors are hard to predict or change with time. So it could prove that this project is a good choice for Finland and Norway.

The next best alternative would be to build a line to Murmansk, which also has a mostly iceless port with good potential connections across the Atlantic and to eastern Asia. This line would be much cheaper to build, because most of it exists or existed, requiring reconstruction of only 200 km vs. more than 450 km to Kirkenes. The advantage of Kirkenes is that currently Norway and Finland are closer neighbors with open borders, so it might be a bit better to use this variant if it can be paid. Even a connection from Rovaniemi to Kirkenes could be built via Russia using existing tracks and requiring only the construction of much less new tracks. Now Russia can be considered to be a reliable neighbor of Finland and Norway, when it comes to such trade routes, but a direct connection that goes only through Norway and Finland might be worth investing a bit more.

Of course Finland does have rail connections to the rest of Europe even today via Sweden and Russia and it does have useful ports in the Baltic Sea coast. They are just not as convenient for long distance trade, especially in winter.


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2019 – Neujahr

Onnellista uutta vuotta! — Καλή Χρονια! — Bonne année! — Frohes neues Jahr! — Un an nou fericit! — Feliĉan novan jaron! — Godt nytt år! — С новым годом! — Срећна нова година! — FELIX SIT ANNUS NOVUS — Gott nytt år! — Щасливого нового року! — ¡Feliz año nuevo! — Gullukkig niuw jaar! — Felice anno nuovo! — عام سعيد — Happy new year!

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Weihnachten 2018

Prettige Kerstdagen! — God Jul! — Crăciun fericit! — クリスマスおめでとう ; メリークリスマス — God Jul! — Feliĉan Kristnaskon! — Hyvää Joulua! — ميلاد مجيد — Срећан Божић! — καλά Χριστούγεννα! — З Рiздвом Христовим! — Natale hilare! — Buon Natale! — Joyeux Noël! — Frohe Weihnachten! — С Рождеством! — Merry Christmas! — ¡Feliz Navidad!

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Hambach Forest


Near Cologne in Germany the Hambach Forest had been occupied by protesters and cleared by the police, in order to promote further destruction of landsacape for the extraction of Lignite. This energy form is just outdated and it is a good idea to get rid of it as soon as possible. Arnold Schwarzenegger brought it to the point in his Video to „Uncle Donald“. What is true for North America, is also true for Europe. Lignite is even worse for the climate than Coal in general and much much worse than natural gas and regenerative energy. All arguments are against using it. But of course the German government has contributed to the heavy overuse of this wrong technology by its forced and hasty nuclear power phase out.

Links (mostly in German)

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New Discussions about Rail Connection from Finland to Kirkenes


Some activity can be observed concerning the question of a railroad connection from northern Finland to Kirkenes in Norway.

Such a railroad would start in Rovanienmi or Kemijärvi and go to Ivalo in northern Finland. From there it would pass the Lake Inari (Inarijärvi) either like todays highway in the west or a bit shorter in the east and lead to Kirkenes at the Russian Norwegian border and the Barents Sea.

Kirkenes has a good port that usually remains ice-less throughout the winter because of the Gulf Stream. It is the eastern end point of the Hurtigruten and there is currently the only road crossing the border from Norway to Russia. Kirkenes can currently only be reached from the land side only by road or by the Kirkenes–Bjørnevatn Line, a railroad that connects the port a a mine just 8.5 kilometers south. It is used mainly for transport of iron ore, depending on the activity of the mine and the world market prices of iron ore.

It seems to be reasonable to expand the port of Kirkenes. The new capacities could be offered to Finland and Russia. Discussions of connecting to Nikel (Никель) in Russia have been slowed down by Russia in an effort to support the port of Murmansk (Мурманск). But the project of connecting to Finland, even though about ten times as far, is being seriously considered. This would allow Finland easier access to the world makets, especially for transatlantic trade. In addition the route will gain importance due to the global warming, which is already making the Northeast passage north of Sibiria a more reasonable route to eastern Asia. It is shorter and cheaper than going via the Suez Canal, but the effect of global warming is still not so strong that this has become the main route. If such a rail connection to Finland existed, it would be possible to use Kirkenes also as port for parts of Russian and Central Europe including the Baltic States both to Asia and the Americas, by just using railroad connections between Finland and Russia that already exist. Finland and Russia both use a broad gauge of 1524 mm or 1520 mm, respectively, that is similar enough to allow usage of both gauges by the same freight trains.

The short line near Kirkenes could be converted to broad gauge or to dual gauge or a new track could be built parallel to it. It would be necessary to build about 550 km of track. Probably road transport could profit also, if the shorter route east of Lake Inari were chosen, because it is likely that a highway parallel to the railroad would close the 20 km long gap between the Finnish and Norwegian highway networks east of Lake Inari.

The shorter connection to Skibotn that was discussed earlier, seems to be irrelevant now. Of course, Skibotn does not have a serious port, which would have to be built, possibly causing some resistance and for sure costing a lot of money. The railroad could also be extended to Tromsø, but that would make it longer anyway and I have not heard of easy expansion options of the port of Tromsø.

Sometimes in Finland it is suggested to build a Tunnel from Finland to Estonia as well to connect to Europe. But I do not see this as a requirement, since the potential of freight transport would anyway come mostly from Finland. Other parts of Europe are already today be connected by train via ferries from Estonia to Finland or without ferries via Russia or via Sweden. One break of gauge will be in the route anyway.

Currently a feasibility study concerning the railroad connection from Finland to Kirkenes is being performed. Sometimes the term „Arctic Ocean Line“ is used for this project. I would assume that we are talking about an electrified single track railroad with short double track sections. The decision is supposed to be made in 2019. This inspires discussions like this article. We will see were this will lead. I am in favor of the project.


Map of likely and possible routes (red):

Arctic Rail Routes

Arctic Rail Routes

Source Wikimedia Commons Creator RicHarc-59 (C) CC-BY-SA-3.0


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Weihnachten 2014

weihnachtsbaum-zuerich-20141218_163548sGod Jul!
Buon Natale!
Joyeux Noël!
Natale hilare!
¡Feliz Navidad!
Merry Christmas!
Frohe Weihnachten!
Prettige Kerstdagen!

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Using Trains in Denmark


The Danish railroads are a sad story for this country. After some experiences as frequent transit traveler on the way from Denmark to Norway or Sweden it is possible to say this without being to much biased by singular experiences that are not representative. The observations all refer to the long distance and international transit transport by rail. The local and commuter train system of the Copenhagen area on Sjaelland seems to be quite OK.

First of all the Danish railroad is an absolute low speed system. It is said that some stretches are prepared for speeds of 140 km/h or even 180 km/h, but real IC trains hardly exceed an average speed of 50 km/h on longer distances, at least when regarding the bird flight distance or at least the shortest highway distance. For a relatively flat country having no elevation that reaches 200 m above sea level and for a country that is not that densely populated this looks like a bad value. I am talking about the connection from Flensburg in Germany to Fredrikshavn and Hirtshals, where no ferries are given reason for longer travel times.

The tracks are very curvy, making them longer than the highway and because of the speed restrictions in curves also slow. Important connections are still not electrified, to be more precise, only the transit railroad from Flensburg to Malmö via Kolding, Odense and Copenhagen, some short sections to Sonderburg and Fredericia and some lines on the main island Seeland are electric. But even fully electrified railroad lines are often used by Diesel trains. On longer trips it is often necessary to change the trains a lot of times, but this changes with every new schedule. The connections are often not so good, forcing waiting times of half an hour. Or if they are good, then they are missed quite often, because connecting trains do not even wait for two minutes on delayed trains with connecting passengers.

It gets especially interesting when trying to take bicycles on the train. The spots need to be reserved. Here some experiences:

  • Reservation cannot be done via Internet or in the railroad station in Germany or Switzerland. The DSB hotline needs to be called.
  • Calling the hotline means listing to music via the phone for at least half an hour. Not expensive these days, but annoying anyway.
  • The hotline girl had created the tickets and wanted them to be picked up at some DSB railroad station of my choice, at least one day prior to my trip. So I would have needed to go to Denmark just to pick up the tickets. Very user friendly approach.
  • Finally I could get the permission to pick up the tickets on the day I intended to travel. The night train from Germany left enough time for that when changing. It was not even eaten up by delay. But nobody knew about my tickets, of course.
  • From counter 1 they sent me to counter 3, from there to counter 2 and from there back to counter 1. There were only three counters, so that was it…
  • In a group of six persons it is necessary to split into different trains, because trains may have enough space for six bicycles, but that is of course forbidden.
  • Taken tandems on the train is forbidden in Denmark. Sometimes it can be done anyway, but it is a matter of luck and friendliness if it is OK or if it leads to a disaster.
  • The hotline person asked for a number to call me back, but expected me to have a Danish phone number.
  • For going from Hamburg to Copenhagen by day train it is necessary to go a 160 km detour via Flensburg, because no train on the direct route transports bicycles.
  • The connection is going only every two hours, so a delay of 25 minutes lead to two hours delay already. The train two hours later did have spare space, but the conductor withdrew her initial offer to use the train without reservation because of the delay. It ended up being only four hours delay, because the German railroad station employee could speak Danish and do the negotiations. In Copenhagen the train was three minutes late and the connecting train would have gone five minutes after the scheduled arrival. Of course they did not wait. There was another train one hour later, but it would only go to Alvesta, not all the way to Kalmar. This train did also more stops, so it was much slower. Going to Alvesta was anyway the plan, so it was OK, but in the end it took 5 1/2 hours more than originally planned. That is 12 hours from Hamburg to Malmö, what would be 300 km bird flight distance. And the German sections are already quite fast from Hamburg to Flensburg or Puttgarden.
  • In the same time it would be possible to do that by bicycle, but transit travel by bicycle is strictly forbidden in Denmark, simply by imposing some short unavoidable bicycle prohibitions on the route.
  • The rides when the air conditioning fails are nice. Especially if there is no water in the washing rooms, so water from the drinking bottle needs to be used for washing the hands. They did provide a tiny bottle of water on the expense of DSB to every traveler on the train.
  • Because of track work there are often buses instead of trains, of course resulting in the loss of all connections. When traveling with a bicycle this even leads to loss of the reservation, see above…
  • The Danish trains are often in a poor state, not compatible with the safety requirements in Germany. So it is resolved by forcing another transfer on the passengers and using the only intact train set as a shuttle for the stretch over the border.

It is also interesting to see the contrast, because Denmark is providing a luxury motorway network providing each village an motorway exit within only a few dozen kilometers distance and this is being expanded at an incredible rate. On the other hand railroad tracks are curvy low speed lines from the budget saving program from the 19th century and the national highways that have no bicycle prohibition are in a poor state and cross every hill and every valley without the slightest attempt to level out the road elevation by dams or cuts. Tunnels or bridges are hardly needed. Is it the intention of Danish transport policy to entirely move all transit traffic from environment friendly means of transportation to cars and air planes? Or are they just lacking the know-how how to build and operate a modern long distance railroad system, but are too proud to get this know-how from countries that have it?

As summary it can be said:
Denmark has no real railroad network, just a few overland tram lines, with speeds typical to the steam power days, but trams usually operate more frequently. This is also what other travelers are telling me. Maybe this is a bit of America in the middle of Europe.

Unfortunately it is not easy to bypass Denmark on the way to Sweden and Norway, but I am constantly looking for such possibilities.

At least in the long term some movement in a better direction might come. They have now announced to improve speed, build some high speed lines and electrify the major long distance lines. We will see if that is just vapor or if it ever becomes reality.


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Why do railroads have no cloverleaf interchanges?


Do cloverleaf interchanges for railroad lines exist?


I must admit that I do not know all railroad lines in the world, but I consider it highly unlikely that anywhere two railroad lines are connected by a cloverleaf. They do exist in games that need not be realistic in every aspect, but this page is not focussed on games or model railroads, so we will ignore this from now on. You might ask the railroad companies why, but at least it is highly plausible. First of all most connections between railroad lines happen in stations. This makes sense for allowing for passengers to change trains and for rearranging freight trains. But there are cases where two rail lines intersect without a major station. Examples that come to my mind are the lines from Darmstadt via Weinheim to Schwetzingen and the line from Mannheim to Heidelberg in Germany that intersect near Mannheim Friedrichsfeld. Or near Zürich were the lines from Wallisellen to Dübendorf and from Stettbach to Dietlikon intersect. The two main lines intersect using a bridge and then some of the connections are made possible using parially level crossings and partially bridges.

The first question is if bridges instead of level crossings are as important for rail as they are for roads. Because the signalling system provides exclusive use of a track to one train, even highspeed tracks may have grade crossings, as long as the switches are built in a way to allow high speed. This does not even slow down the high speed trains during normal operation. The highspeed train gets the track and all other trains have to wait. Schedules are usually made in a way that minimizes such waiting times, so they usually only occur to greater extent if some train is delayed. But if you look how railroad infrastructure has been built it will be obvious that bridges between different tracks are quite common, at least in busy sections of the network. Other than for road transport this is less an issue of speed and security and more an issue of capacity. The bridges allow for more simultaneous train moves and there are less situation in which one train blocks many tracks at the same time mostly near a station. This has been known in the 19th century and a typical construction for a station connecting two rail lines included a bridge for crossing over between the two lines coming from one end. Assuming lines coming from north, east, south and west and a station that is oriented from north west to south east, the lines from the north and the west enter on one side and those from the south and the east from the other side. With this bridge between the lines coming from north and from east the typical setup of trains going from north to south and from east to west allows all four directions to pass simultaneously if the lines are double track.

Typically level crossings are only avoided for combinations that are used frequently. Blocking some tracks by rare operations which might even happen during night times is not a priority.

But returning to the cloverleafs: They are not really grade separated and they would use tremendous amounts of space because of the minimal curve radius of railroad tracks.

Here is such a cloverleaf which has in a way level crossings („weaving“) in the red areas:
With low traffic densities and traffic consisting of small individual vehicles like bicycles, cars or trucks this works just fine. And a highway that seems to be filled with cars does not really have a very high traffic density because cars are so wasteful with space. As long as the traffic of two neighboring leaves combined is less than the capacity of one lane this can work. Otherwise even for highways other constructions that are really grade free and less space consuming, but more expensive to build seem to be more adequate. Some of them could actually work for railroads as well, because they widely avoid unnecessary curves.

Very elegant because it can be built with two levels and is anyway fully grade free. But not good for railroad tracks because of the 270°-bows. This does not work very well either for more than one roadway per direction.

This works for any number of tracks, road ways, bicycle ways, tram tracks, bus lanes and whatever combined by just adding more curves. But it needs four levels instead of two. The same principle works for combining n throroughfares meeting in one point using n levels for
n=3, 4, 5, 6,.... At some point these interchanges grow so high and the ramps for the height differences so long that it would no longer be possible to build this, even with a lot of money. It is very unusual to see such constructions for n>4.

Here are some more setups:





There are much more possibilities. Hopefully we won’t end up with something like this:

It is absolutely possible to connect any number of throughfares with any number of tracks, roadways bicycle lanes, bus lanes etc. avoiding 270°-bows and using just two levels in a fully grade free way. The idea is very simple and I just describe it for tracks for the sake of simplicity: All tracks have only one direction. The lower level carries tracks running from north to south and vice versa in the vicinity of the interchange, possibly turning wherever they really go. The upper level has tracks running from east to west and vice versa. Each incoming track splits into two tracks. On of them continues to the other side and only takes incoming connections. This is a collector. The other one only branches into curves connecting to the other tracks, called distributor. All tracks cross the field, using some space between different tracks. Whenever a collector and a distributor cross, the appropriate connection is present. After having crossed the whole field, for example from south to north, a distributor splits into two distributors going east and west until the last collector has been met. This allows for all possible connections in a fully grade free way and with no capacity constraints other than those imposed by the tracks leading and from the interchange. It needs very much space. And it is a theoretical construction, because in practice it seems to be sufficient to offer the most important connection in a grade free way. Real railroad interchanges are relatively compact, but the station or the yard that happens to be located at the interchange can be huge.

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