Beautiful Shapes

I could have named this blog Uniform Polyhedrons, but I think in that case not many would have read it 😉 A polyhedron is a 3D object, bounded by polygons and a polygon is a flat surface, boudned by straight lines. A cube is a simple polyhedron and a triangle is a simple polygon.MOre terminology in the appendix.

When I was a kid, I was fond of making cardboard models of buildings, ships etc. I bought the “bouwplaten” in the local bookstore. It was quite a popular pastime in those days, now no more. Here are two simple examples, found on the Internet.

It was during the 1970s , on a trip to London, that I came across the book Polyhedron models by Magnus Wenninger. It contained descriptions of 119 polyhedrons with detailed instructions how to make cardboard models of them. With my youthly love of bouwplaten and my interest in mathematics I immediately bought the book. Left my copy, right Magnus Wenninger (1919-2017) with a complicated polyhedron in his hands.

Back home, I bought sheets of colored cardboard and started building polyhedrons. Compared with the commercial “bouwplaten” as shown above, where you just have to cut out he various pieces, you have to draw the pieces first on the cardboard sheet, add tabs and then only cut them out. Here are two examples. The numbers are from Wenninger’s book, which can be found online.

The tetrahedron (left)is the most simple polyhedron, it consists of just four triangles. I have marked how many pieces you have to cut with a colored number. The football like polyhedron with the unspeakable name (right) consists of 30 squares, 20 hexagons and 12 decagons. 62 pieces in total.

Here are a few of the polyhedrons I have built. That was more than 40 years ago, the colors have faded. The polyhedron in the center of the top row is still “simple”, consisting of squares and triangles. The one left on the top row looks more complicated, but when you look carefully, you will see that it only consists of triangles! But only parts of a triangle are visible from the outside. In the right polyhedron, on the bottom row it is easy to see that there are pentagons (five-sided polygon), but there are also hexagons (six-sided polygons), which are hardly visible in this model. In total 12 pentagons and 10 hexagons!.

The polyhedrons where all faces are completely visible, are called convex, the others where you can only see parts of the faces are called nonconvex. See the appendix for more terminology and mathematical details.

Nonconvex polyhedrons are more difficult to build, because you have to be careful that the pieces of one polygon have the same color. But they are worth building, because they are beautiful. Here are a few examples. The left polyhedron consists of 12 pentagons and 12 pentagrams, 24 faces in total. The one at the right is more complicated , 20 triangles, 12 pentagrams and 12 decagons (10-sided polygon), total 44 faces.

Two more. The polyhedron left has 30 squares, 12 pentagons and 12 decagons, total 54 faces. And the beautiful polyhedron to the right has 20 triangles, 30 squares and 12 pentagrams, total 62 faces. The complexity of this polyhedron is difficult to see in a picture. On Wikipedia I found a 3D version which you can rotate with your mouse. Amazing, try it out and see if you can find the triangles (easy) and the squares (difficult).

The polyhedrons at the end of Wenninger’s book are even more complex, Here is a description with templates for the “Great Inverted Retrosnub Icosidodecahedron“. Yes, they all have names, see the appendix. It contains 80 triangles and 12 pentagrams, 92 faces in total .His description starts with “This polyhedron is truly remarkable in its complexity” and at the end he writes “Your patience and perseverance will have to hold out for more than 100 hours if you want a complete model of your own

At first I decided that “more than 100 hours” was too much for me. But I was curious about this polyhedron, and I used the templates to build a small part of it.. Soon I found out that there was something wrong with the templates for this model. Parts that had to be glued together, had different lengths! I tried to check and correct the size of the pieces (see right image with my comments) but that did not work..

I decided to contact Wenninger, but didn’t have his address, so I wrote to the Cambridge University press ( the publisher), asking them to forward my letter to Wenninger. I didn’t really expect a reply, so I was pleasantly surprised when after a couple of months I got a letter from Wenninger. He explained that in the printing process of the book one or two templates had been incorrectly represented. A few more buyers of the book had noticed the error. His letter contained the correct templates!.

After his kind gesture I felt “morally” obliged to build the polyhedron. I spent many evenings cutting and gluing the 1290(!) pieces. I did not keep track of the hours, but it must have been more than 100. Here is the final result. Of course I took a picture and sent it to Wenninger.

Here is a digital 3D version of the polyhedron. Rotate it with your mouse, to see the complexity.

I assume that in a reprint of the book the mistakes will have been corrected, but when I built the model, it must have been one of the few in the world ;-). Years later I visited the Science Museum in London, where they have the whole collection.

Polyhedrons have fascinated artists, philosophers and mathematicians throughout the ages. Here are Durer;s famous Melencolia I (1514) and John Cornu’s Melencolia (2011)

Appendix

First some terminology.

  • A polygon is a 2D figure with straight sides, for example a triangle. When all sides are equal it is called a regular polygon
  • A polyhedron is a 3D form bounded by polygons, for example a cube. A polyhedron has faces, edges and vertices (plural of vertex) When the polygons are regular and all vertices similar, the polyhedron is called uniform.

The left polyhedron has 6 faces (F=6), 12 edges (E=12) and 8 vertices (V=8). The right polyhedron has F=4, E=6 and V=4.

The most simple polyhedrons were already known in antiquity and are called Platonic solids. These polyhedrons have only one regular polygon as face. , a triangle, square or pentagon. Here they are

There are 13 polyhedrons that have more than more than one regular polygon as face.. They are called Archimedean solids, because they were first enumerated by Archimedes, later rediscovered by Kepler who gave them their names. Here they are. Notice that they all have one single edge.

The names give information about the composition of the polyhedron. For example the icosidodecahedron has 20 (icosi) triangles and 12 (dodeca) pentagons.

The polyhedrons often contain pentagrams. A pentagram is related to a pentagon by a process called stellation, extending the sides of a polygon. Polyhedrons can also be stellated by extending their faces. Left the pentagram and right one of the stellated dodecahedrons.(there are three more)

In the Platonic and Archimedean polyhedrons all faces are completely visible, The mathematical term is that these polyhedrons are convex. The stellated dodecahedron, shown above, has pentagrams as faces, but the center part of the pentagram is not visible, it is inside the polyhedron. The mathematical term is that this polyhedron is nonconvex. In total 53 nonconvex polyhedrons exist. This has been proven only in 1970.

Wenninger’s book describes 119 uniform polyhedrons, the 5 platonic solids, the 13 Archimedean ones, 48 polyhedron stellations and the 53 nonconvex polyhedrons. A List of Wenninger polyhedron models can be found on Wikipedia. The list contains images of all polyhedrons and lots of details

Here are the numbers of the polyhedrons shown in this blog (I have built more). 17, 24, 39, 76. 80, 99, 102, 105, 107 and 117.Except 39, a stellation of the icosahedron, they all have a Wikipedia page.

When I built my models, PC’s were still in an infant stage and the World Wide Web did not yet exist. Nowadays there is wealth of information available, there even exists software to create the polyhedrons digitally. Great Stella looks promising. I feel tempted 😉

Why did I write this blog, more than forty year later? Recently I visited the Bellevue Hotel in Penang. The owner of the hotel is a friend of mine. In the garden of the hotel he has built a geodesic dome. He was a close friend of the American architect and philosopher Buckminster Fuller (1895-1983), who was the “inventor” of the geodesic dome.

You will not be surprised that there is a close relation with the polyhedron models of Magnus Wenninger. Have a look at the Wikipedia article Geodesic polyhedron, where both Buckminster Fuller and Wenninger are mentioned. Enjoying the view and admiring the dome, the thought arose to write a blog about my “hobby” from the past 😉

Batang Kali again

Half a year ago I published a blog My Waterfalls in which I described 48 waterfalls visited by me (with my friends) since 2007, but never included in my website Waterfalls of Malaysia. When you look at the list, you will notice that there has not been much waterfall activity during the last few years. After 2017 only two “new” waterfalls in 2021, the Batang Kali fall and the Pencheras fall. The reason for my lack of activity was partly the COVID-19 pandemic, limiting traveling possibilities, but mainly my growing lack of confidence.

In 2015, during a rewarding “expedition” , Sg Siput Waterfall Recce, we had to cross a river to reach the intimidating Lata Kaku.

In my blog I wrote: I don’t know why, but I felt uneasy, stumbling often, maybe the years are counting…. In the following years that feeling got stronger. I have decided not to visit remote waterfalls anymore. And also only hike in the jungle when accompanied by at least two “strong” “young” friends ;-). My visit to the Batang Kali waterfall in March 2021 is a good example. I went there with Edwin and Teoh, the hike took about one hour and the waterfall, though not spectacular, was nice with a big pool.

When I told my Dutch friend Paul about this waterfall, he was interested, so we needed two young men to join us. Fahmi was willing, but Aric was too busy with his laundry shop. Fortunately Rani was available. On 23 July we met at the Kedongdong recreational park, from where it was a 800 meter walk along. the tar road to the trailhead.

Right after the start we had to cross the Pencheras river. Last year we had to wade through the river or cross on a fallen tree, now a simple bamboo bridge had been built.

There was a trail, although not always clear. Locals collect bamboo here, there were remains of sheds and several side trails, but with the help of Rani we found our way.

Many bamboo groves.

Here and there fallen trees blocked the trail.

Halfway we had to cross the Batang Kali river to the other side. No bridge here, so we got our shoes wet.

There was still a trail and of course there were numerous leeches. Tiger leeches mainly.

After about one hour we arrived at the waterfall. More water than last year.

Enjoying a bath.

Of course we took many pictures.

Upstream there are more waterfalls, but they require river trekking. A few years ago I would have continued to explore them. Now this waterfall was enough for me. After playing around for a while we hiked back. Time enough to look around and enjoy nature.

Almost back, we came across a few spectacular ginger plants. Left a torch ginger, right a beehive ginger.

After the hike it was time for lunch. Rani had to hurry back for a birthday party, Paul, Fahmi and I went to the World of Phalaenopsis café in Ulu Yam. Pleasant surroundings, nice food.

In the future I hope to have more waterfall hikes, similar to this one.


In one of my blogs I mentioned the Australian collection of Malaysian topo maps. I was interested to use one of these maps and add my waterfall trips. Left the map, published in the 1940s, notice how few roads existed in those days. Estates everywhere. Right the same map, with my trips indicated. The Batang Kali waterfall is at the right .

BepiColombo

A few weeks ago I read this in the news:

BepiColombo Spacecraft Makes Second Gravity Assist of Planet Mercury – Captures Spectacular Close-Ups

Here is one of those close-ups.

The BepiColombo spacecraft? I am interested in space missions and have written several blogs about space travel and spacecrafts, but I must have missed this one.

So here is a post about BepiColombo. And about Mercury. And about Gravity Assists.

Let me start with Mercury, the smallest of the eight planets in our solar system. And the fastest, orbiting the Sun in 88 days. Its orbit is the most elliptical of all planets, the distance to the Sun varies between 46 and 70 million km. (For comparison, the similar distances for Earth are 147 and 152 million km).

Mercury is not easy to observe from Earth, because the planet orbits so close to the Sun. For a long time, it was thought that Mercury was tidally locked to the Sun, in the same way as the Moon is tidally locked to Earth. It was only in 1965 that radar observations of Mercury showed that it was actually rotating with a period of 59 days. An Italian scientist, Giuseppe Colombo noticed that this value is 2/3 of the orbital period and suggested that Mercury and the Sun are in a so-called 2:3 resonance, with Mercury rotating 3 times during 2 orbital periods. More about tidal locking and resonances in the appendix.

In the nineteen sixties space travel started, in the USA with the Mariner program from 1962 to 1973. Here are a few of the Mariners. The Mariner 2 was the first spacecraft to reach another planet (Venus), It had not yet a camera on board! The Mariner 4 flew by Mars and took 20(!) pictures of the red planet. .

The Mariner 10 mission had a novelty, after its launch it passed very close to the planet Venus. The gravitation of this planet changed the speed and direction of the Mariner in such a way that it continued its course in the direction of Mercury. This is called a gravity assist, often (confusingly) called a gravitational slingshot. See the appendix for more details.

.In the left diagram you see the effect. Three months after launch the Mariner 10 passes Venus at a distance of less than 6000 km. It brings the spacecraft in an elliptical orbit around the Sun with a period of 176 days. On 29 March it passes Mercury at a distance of 700 km. For the first time in history pictures were taken of Mercury’s surface!, A big surprise was that Mercury had a (weak) magnetic field, so it should have a liquid iron core.

The gravity assist was suggested by the same Giuseppe Colombo and was so successful that it is now a standard procedure for spaceflight.

It took almost 30 years before the next mission to Mercury started. In 2004 the MESSENGER spacecraft was launched and its mission was to go into orbit around Mercury and study its structure and magnetic field. Going into orbit around Mercury is not an easy job because of the strong pull of the Sun. Not less than seven gravity assists were needed to slow down the spacecraft enough, one flyby with Earth itself (!), two with Venus and four with Mercury. Here is a diagram of the flight path. Just to show how complicated it is.

The advantage of gravity assists is that you don’t need fuel to change the course, only minor DSM’s (Deep Space Maneuvers). The “disadvantage” is that it takes considerably more time to reach the target. In this case more than six years.

After this lengthy introduction, let’s go back to the BepiColombo mission. Giuseppe (Bepi) Colombo died in 1984, this mission must have been named BepiColombo in his honor, as he was the first to identify the 2:3 resonance of Mercury and also the first to suggest a gravity assist for the Mariner 10 to reach Mercury..

BepiColombo is a joint mission of the European Space Agency (ESA) and the Japan Aerospace Exploration Agency (JAXA). BepiColombo was launched in October 2018. The spacecraft contains two orbiters, one MMO) to study the magnetic fields of Mercury, the other one MPO) will study structure and geology of the planet.

In this animation, you can follow the flight path of BepiColombo (pink) from the launch in 2018 until it goes into orbit around Mercury in 2025. The orbits of Earth, Venus and Mercury are in dark blue, light blue and green, respectively. The spacecraft will use a total of nine(!) gravity assists before it goes into orbit.

As it may be difficult to see where and when the flybys occur, I have taken a few screenshots from a very informative video created by ESA: BepiColombo – orbit and timeline .Worth watching. In the screenshots the flyby is indicted with a circle.

The photo of Mercury at the begining of thos post was, taken during the 2nd flyby of Mercury on 23 June 2022.

When BepiColombo goes into orbit around Mercury, it will have travelled more than 10 billion km. Only then it will deploy the two orbiters.

So we will have to wait more than three years before the two orbiters start collecting scientific data.

Appendix: Tidal locking

As probably everybody knows about tides on earth, we will start there. Twice a day the sea will have a high tide and a low tide. Those tides are caused by the gravitational attraction between Earth and Moon. This force depends on the distance between the two bodies. It is a bit stronger on the side of the earth facing the moon, than on the opposite side, resulting in the tides.

The friction caused by these tidal forces, will slow down the rotation of the Earth, increasing the length of a day. Not much, about 2 milliseconds per century. But when Earth and Moon were formed, about 4.5 billion year ago, the length of a day was much shorter only a few hours.

A similar story holds for the Moon, but here the slowing down has been so effective that for billions of years the moon is “tidally locked”, the rotation if the moon (its “spin) is equal to its orbital period around Earth. The technical term is that the Moon is in a 1:1 spin-orbit resonance with Earth. From Earth we always see the same side of the Moon.

Most other moons in our Solar System are also tidally locked to their planet. For example the four Jupiter moons, discovered by Galileo in 1609.

An interesting case is Pluto (no longer a planet) and its moon Charon. Charon is a large moon and Pluto a small “minor planet”.. Both moon and planet are tidally locked to each other! Here is an animation.

The gravitation of the Sun aldo causes tidal forces on the planets. On Earth we are aware of that, but the Sun’s tidal forces are smaller than those of the moon. During full moon and new moon the two tides enhance each other, the high tide is stronger and called a spring tide. During first and last quarter they work against each other, the high tide is weaker and called a neap tide. See the diagram below

Because Mercury is orbiting so close to the Sun, the tidal forces are a lot stronger. Until 1965 it was thought that Mercury was tidally locked to the Sun, rotating in 88 days, same as the period of its orbit => a 1:1 resonance. Now we know that it is a 2:3 resonance, Mercury rotates faster, 1.5 times during one orbit. The reason is that Mercury’s orbit is quite elliptical, so its (orbital) speed is not constant, moving faster when it is close to the Sun. Here is link to a good explanation: Mercury’s 3:2 Spin-Orbit Resonance. .

The length of a day is commonly defined as the time between successive sunrises or sunsets. 24 hours for Earth, slightly more then the rotation period of 23.9344696 h. With 1:1 tidal locking there is no more sunset/sunrise, the concept of a day has no meaning or you could say that the length of a day is infinite ;-). The animation below shows Mercury orbiting the Sun. The red point represents an observer on Mercury. Note that this observer rotates three times during two orbits. Dawn, midday, dusk and midnight are marked. A day on Mercury takes 176 (earth) days, much longer than the rotation period of 59 (earth) days!

Appendix: Gravity Assists

After launch, a spacecraft will move under the influence of gravitation, primarily the attraction of the Sun. Using the precious fuel on board, it can maneuver a bit to reach its destination. When its course brings it close to a planet, the gravity of this planet can change direction and speed of the spacecraft, without using fuel. Depending on how the spacecraft approaches the planet, its speed can increase or decrease. This use of a planet’s gravity is called a gravity assist or a gravitational slingshot.

Here is a somewhat misleading analogy of a gravity assist. “Space balls” are shot at a train with speed of 30 MPH. If the train is at rest, they bounce back with a speed of 30 MPH. But the train is not at rest, it approaches with a speed of 50 MPH. The balls hit the train now with 30 + 50 = 80 MPH and bounce back with the same speed. For the observer along the rails, the balls now have a speed of 80 + 50 = 130 MPH.

This analogy, from Charley Kohlhase, an important NASA engineer, illustrates a few important points. 1).The balls are interacting with a moving object and 2). the mass of the moving object is so large, that its loss of energy can be neglected.

My own favorite example is that of a tennis player, who hits an incoming ball, before it bounces (a volley). When he keeps his racket still, the ball will bounce back with (about) the same speed (block volley). When he moves his racket forward, the speed will be larger (punch volley), when he moves it backwards, the ball will go back slower (drop volley). In this case his own mass is less than the train, so he will feel the impact of the ball.

In space there are no contact forces, everything moves under the influence of gravity, therefore I always found the analogy unsatisfactory. The influence of gravity on the motion of two bodies in space has been described by Kepler using Newton’s gravitation law. We assume that the mass of one body (a planet) is much larger than the mass of the other one (a spacecraft) Here are a few possible orbits. The red one is part of an ellipse, the green one a parabola and the blue one a hyperbola.

On the Internet you can find numerous videos explaining gravity assist. Pick your choice here. Many of them I found confusing and/or too complicated. So I decided to give it a try myself ;-). Here are three images I have created.

The left image shows the course of a spacecraft under the influence of a planet’s gravitation. It is a hyperbolic orbit, where the speed increases until the spacecraft is closest to the planet (called the periapsis), after which its speed will decrease again. The initial speed and the final one are equal, only the direction has changed (the red arrows). If the planet would be at rest relative to an observer (for example Earth), that would be all.

But that is not the case, the planets move around the Sun. In the second image, a planet moves to the right (blue arrow). The gravitation between spacecraft and planet is still the same (the red arrows) but an outside observer will now see the effect of the two speeds: the green arrows. The change of direction of the red arrows now has a clear effect, the final speed is larger than the initial one: here we have a gravity assist to increase the speed of the spacecraft!. This happens when the spacecraft passes “behind” the planet.

In the last image I have reversed the speed of the planet, so now the spacecraft passes “in front of” the planet. With an opposite effect, now the final speed is less than the initial one, The gravity assist in this case reduces the speed of the spacecraft.

Spacecraft exploring the outer planets have to overcome the gravitation of the sun and will need an “extra push” from gravity assists, passing at the rear of planets. BepiColombo is getting closer to the Sun and has to break to be able to go into orbit around Mercury. Therefore it needs gravity assists, passing in front of a planet, reducing its speed.

For me, this explanation of a gravity assist is satisfactory, I am curious about the opinion of others. Comments are welcome 😉