While the others fend off the zombies, Richtofen pretends to use the Summoning Key to acquire the original Richtofen's soul, which Dempsey notices. Eventually, the four activate a beacon in the facility, allowing Maxis from another dimension to locate them. The Germans had captured Ultimis Dempsey's cryo chamber from being experimented on by Ultimis Richtofen and planned to bring him to Griffin Station, now fully ran by Dr. Groph in Ultimis Richtofen's "absence", with Primis close behind until a German soldier fires a Panzerschreck at the Giant Robot, disabling it before it could retrieve Ultimis Dempsey.

After arriving at Griffin Castle, the young four watch as a rocket containing the Ultimis Dempsey launches to the Moon. After making radio contact with Dr. Groph, Richtofen masquerades as his Ultimis, older self in order to fool Groph. However, Groph becomes suspicious of Richtofen's change of personality and announces him as an imposter. Using the Death Ray, the young four manage to bring the rocket containing the Ultimis Dempsey back to the castle, much to Groph's anger, who activates a failsafe so they couldn't touch the chamber.

Using the Vril Device obtaining from the crash site, the young four awaken a ghost Keeper and help it return it to its physical form before it returns the favor and brings the M. However, the Keeper becomes corrupted, forcing the four to battle it, eventually defeating it. After defeating it, Groph expresses his extreme anger, vowing to destroy them and the castle itself. In order to tie up these loose ends, the four use the Summoning Key to launch several missiles towards the Moon, destroying it, ultimately destroying Griffin Station and killing any Group scientist at the station, including Groph.

After walking towards Ultimis Dempsey's cryo chamber, Richtofen activates the Summoning Key, forcing the others to levitate in the air. Here, Richtofen's plan is revealed to the others. Richtofen plans to kill the Ultimis versions of themselves to be put in the Summoning Key like he supposedly did with his own at the Der Riese facility.

When Dempsey himself sadly put his older self down, Richtofen briefly comforted him after it was done, showing that he had laid his previous dislike of Dempsey off to the side. Richtofen then captures the Ultimis Dempsey's soul in the Key. Richtofen and the other three then travel to a different fractured timeline, to an island, arriving on a Japanese vessel heading for a Division 9 Facility on October 18th, , before being caught and the Summoning key being taken away from him.

A brief fight ensues with the Japanese soldiers with Richtofen burning one alive before he notices the Key rolling toward the ocean, luckily Takeo had managed to barely catch the Key by his fingertips much to Richtofen's relief. The four then are forced to swim to a nearby island after the ship blows up. Later, Richtofen was present when the Ultimis Takeo Masaki was cured from being a Thrasher to collect his soul. After the Ultimis Takeo honorably sacrificed himself for a better future, Richtofen then preserved his soul. Initially Dempsey suggested going after the Ultimis Nikolai but Richtofen said a "chain of events " must be set in motion.

Realizing that he wants to save his three comrades from their eventual fate, Richtofen then takes them to Alcatraz on July 4th, , where he meets his younger self and acquires the Victis blood samples. Primis then jump to another fractured timeline, in a war-torn Stalingrad on November 6th, , to kill the Ultimis Nikolai and place his soul in the Summoning Key. Eventually the four are teleported out of the sky and parachute in front of Ultimis Nikolai's mech while he is distracted with vodka and listening to "The Ace of Spades" on his radio.

Upon seeing them the Ultimis Nikolai attempts to kill them until a dragon throws him into a building. Later, Richtofen and the others make a temporary truce with the Ultimis Nikolai in order to kill the dragon. Afterward, Richtofen and the other three demand Ultimis Nikolai to surrender to which he refuses and betrays them.

After this, he calls to Maxis, telling him that they'll be meeting him soon. Soon after the events of Gorod Krovi, Richtofen along with the rest of the Primis crew finally arrive at the house. There Richtofen reunites with Maxis, the two then destroy the MDT to ensure the house would remain safe. Letting his guard down Richtofen leaves the Soul key on a table unaware of who was trapped within. The voice inside calls out to Maxis and influences him due to his lack of soul making him easy to control.

A , Primis manages to kill the Shadowman and banish the Apothicons to places unknown. As the world returns to normal Monty becomes concerned as Primis remains in front of him. Noticing the empty blood vials he becomes angered claiming that they are putting his realm at risk. Initially he decides to simply wipe them from existence but decides to instead send them somewhere else. The 4 then end up in the year , in northern France just as the Great war comes to an end causing the cycle to continue.

Set before the events of Gorod Krovi, Primis enters the pocket dimension of Alcatraz to pick up the blood vials. Ending up far from their intended target, the group enters the underground lab and find another Richtofen. He gives the vials to the group and instructs Richtofen to re-read the Kronorium. Richtofen scans through pages exclaiming that pages have changed and that his blood is needed. He becomes enraged and throws the Kronorium before hearing a horde of zombies.

The group then makes their escape but before entering the portal Brutus destroys it trapping the group within the twisted version of Alctraz. Beside a mysterious machine known as the Dark Mechanism, The Primis crew battle against Brutus, Whom is being resurrected time and time again. Richtofen realized what the Kronorium meant and that the machine was intended to draw his blood, and stepped inside, sacrificing himself to free the others.

With Brutus finally unable to resurrect, the rest of Primis is joined by a post-Revelations Richtofen, who informed them that the future had changed, and that Nikolai had to keep his soul to defeat Doctor Monty. The future Richtofen explained that the cycle was broken, and that he wished he could say he was sorry to his younger self. The remainder of the Primis members and the original Dimention 63 Richtofen, leaving the new Richtofen alone and scared, the machine continuing to drain his blood.

In order to prepare for the coming battle; Primis makes a stop at Groom Lake to gain some allies. Within Hanger 4 the Ultimis crew were being held seemingly after the events of Moon, With Richtofen also retaining his body. As Ultimis Nikolai expresses glee at being able to drink uninterrupted, Primis Nikolai emerged from a portal asking his alternate self if drowning his sorrows in vodka is the best choice.

The Primis crew rallies their alternate selves and recruit them to fight in the coming battle. A Great war. Richtofen is a psychopath with a desire for violence, often laughing maniacally when killing. He loves blood and death and thoroughly enjoys killing zombies, and has a strange obsession for the spleen. He also has a strange suggestive feeling towards the zombies when killing them. Despite his insanity, he retains his knowledge of bodily organs, technology and Element It's been noted when he is not killing the undead, he is on the calmer, quieter side, talking to himself alone quite a bit.

He is also adamant on completing his work quickly. He despises Dempsey, likes Nikolai and has mutual feelings towards Takeo. He is a megalomaniac due to his plans on destroying the world with a zombie army which partly fails due to intervention from Ludvig Maxis. Post Moon he retains his insanity as the Demonic Announcer. After the Earth was destroyed he became focused in mending the Rift to eternally damn Samantha in the Aether world and to destroy Maxis once and for all as well as to partially repair the damaged Earth.

Due to him still being sane at this point, Richtofen's personality is completely different. He appears level-headed but paranoid possibly due to exposure to Element in France.

## A comparative look at a centenary. A review of World War I commemorative events in Brussels

He appears more unskilled due to only being a field scientist and does not go into battle often, but attempts to keep focused nevertheless. He appears more fearful of the zombies and is more cautious for his own survival. After some time staying in the House, Richtofen becomes more cheerful as evident when greeting Victis. However, throughout his journey in acquiring the souls for the Summoning Key, Richtofen takes on a more serious, grim personality, rarely ever smiling or cracking jokes.

His sudden acquisition of knowledge from the Kronorium, along with repeated dimensional traveling, also made him mentally unstable at times, repeating phrases throughout and misremembering events. This goes for all other characters in the Primis group as well, as their personalities are much more like their Ultimis counterparts when compared to their first appearances in the map, "Origins". Sign In Don't have an account? Start a Wiki. Contents [ show ]. Shi No Numa. Der Riese. Kino der Toten. Call of the Dead Heard only.

Green Run As Demonic Announcer. Die Rise As Demonic Announcer. Buried As Demonic Announcer, if Mined Games completed in his favor joins Stuhlinger's body, becomes zombie if completed in Maxis' favor. Origins Primis version Ultimis version appears as Eddie in the outro cutscene. Shadows of Evil Primis version Appears in easter egg end cutscene Ultimis version appears zombified as a jumpscare. The Giant Primis version Ultimis version dies in cutscene. Der Eisendrache Primis version. Zetsubou No Shima Primis version. Gorod Krovi Primis version.

Revelations Primis version, Ultimis version's soul appears as Eddie in the intro cutscene. Blood of the Dead Primis version, dies in ending cutscene. Classified Ultimis version, Primis version appears in cutscene. Richtofen holding a Ray Gun at Der Riese. Richtofen in Call of Duty: Zombies. Kino der Toten 's portrait of Richtofen in the original Black Ops. Richtofen holding a Ray Gun at Kino der Toten.

Richtofen using the JGb Richtofen in Moon. Gamer Picture that is received after completing the Eclipse Easter Egg only. Richtofen's character model, as seen in Samantha's room in Kino der Toten. Richtofen, with other characters, as seen in Origins. The child version of Edward Richtofen seen in the House from the Origins quest ending.

Edward Richtofen in Shadows of Evil. The jumpscare found in Shadows of Evil. Richtofen shocked and horrified by Division 9's experiments. This ME, all of it will be gone. Richtofen collecting the Ultimis Nikolai's soul in the background. Primis with the Shadowman in the background. Monty examines Richtofen's empty blood vial.

Primis Richtofen wielding the Staff of Wind. Promotional art for Zombies Chronicles featuring Ultimis Richtofen. Ultimis Richtofen in Zombies Chronicles. Future Primis Richtofen meets with his past self in the intro cinematic of Blood of the Dead. A new portrait of Ultimis Richtofen in Classified. Heinrich Amsel. John F. Mullah Rahman. Jonas Savimbi. The Replacer. Categories :. Cancel Save.

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VC Bookie. Joint Special Operations Command. United States Secret Service. Tian Zhao. Panamanian Defense Forces. In the case of icosahedral symmetry, we have a full characterization of when they occur: As described by Dutour and Deza, every fullerene of I h or I symmetry is a Goldberg-Coxeter transform of C It is also possible to determine when these are of I h and when they are of I symmetry.

The volume and surface are of a regular sided polyhedron icosahedron, see Figure 3 made out of equilateral triangles with edge length a is,. This gives the ratios and between the two volumes and surface areas. A smaller bond distance in the hexagon implies that more of the original icosahedron is cut off,.

For C 20 equal edge lengths the vertices lie in the center of each face of an icosahedron, and the volume and surface area are easily obtained,. How can we get the volume V and surface area A for any fullerene isomer? As fullerene cages are not guaranteed to have planar faces, their volume or surface area are only approximately defined.

There are, however, a number of definitions according to which we can express both quantities. We could triangulate all faces by adding a barycenter to each face with lines to each vertex of that face. We call the polyhedron obtained in this way a triangulated face polyhedron , TFP. For a fullerene, the graph representing the TFP is identical to the dual structure of its leap-frog transform. The total surface area A is obtained by summing over all areas A i of the triangles obtained, which works even for faces where the vertices do not lie on a plane. Using Gauss' theorem divergence theorem , the volume V is found by summing over the face normals,.

The different volumes and surfaces areas can be used for calculating some important measures for fullerenes, such as the sphericity S how spherical a fullerene is and convexity C how convex a fullerene is. The surface of a triangulated fullerene may not be convex. We can measure the non-convexity by comparing to the convex hull of the fullerene cage, which is the smallest convex polyhedron that contains all the points.

The convex hull CH is uniquely defined, and there are several algorithms available, such as the incremental 3D convex hull algorithm. The simplest measure of sphericity is the isoperimetric quotient q IPQ , defined for a polyhedron as. D IPQ is shown for several fullerenes in Figure Various deformation parameters D in percent for a series of fullerenes selected according to stability. For larger fullerenes, Goldberg-Coxeter transformed structures of C 20 were chosen.

Geometries were obtained from DFT up to C or force field optimizations. We may therefore analyze whether or not the vertices of a given fullerene lie on a sphere. For this, we define the minimum covering sphere as a sphere of minimum radius that encloses all vertices in the polyhedral embedding J. Sylvester, The ellipsoidal problem has been addressed recently, and is known as the minimum volume axis-aligned ellipsoid problem MVAE. We can now define a number of useful measures for sphericity.

The MCS definition for the distortion is biased to the case of few atoms sticking out on a sphere and another measure may therefore be more appropriate. We define the minimum distance sphere as. A similar definition that uses the mean deviation from the average distance taken from the barycentric point has been introduced by Nasu et al.

Figure 4 shows a comparison of sphericity parameters for a number of stable fullerenes. For C 60 , all vertices lie on a sphere and all deformation parameters are zero, except for the IPQ, which except for its simple definition is perhaps not the best measure for sphericity. All deformation parameters reveal the highly deformed fullerenes spikes in Figure 4. In the solid state, the interactions between the fullerenes are of Van der Waals type. Note that the polarizability and therefore the Van der Waals coefficients grow linearly with increasing number of carbon atoms in the fullerene cage, which should converge toward the graphene limit for the per-atom value.

Experimentally, the solid state behavior of C 60 has been studied in great detail. However, once the Van der Waals space is squeezed out and the fullerene cages touch, the bulk modulus increases substantially, as the fullerene cage is not easily compressible, similar to the high in-plane stiffness of graphite or graphene. Figure 15 shows lattice constants for a number of fullerenes using this hard sphere model, which can be taken as an upper bound to experimental lattice constants.

For the other experimentally known fullerene crystal structure, C 70 , the fcc lattice constant a fcc of The dependence of the a fcc lattice constant on the vertex number can be expressed as , where the shift follows from the geometry of the fcc cell and the -term represents the dependence of the fullerene radius on N. For distorted fullerenes, the packing problem becomes far more complicated.

For example, we could add the Van der Waals radius to every carbon atom and take the convex hull around this Van der Waals layer. We are then faced with close packing of complicated polyhedral structures, which is a difficult, unsolved problem. However, there exist algorithms for packing arbitrary polyhedra in finite spaces, , and it is possible that one could approximate properties of the periodic close packing solution by making the finite cell large enough.

For some fullerenes, ellipsoidal, or cylindrical covers are more appropriate as already mentioned. This is related to one of Hilbert's fundamental problems: How can one arrange most densely in space an infinite number of equal solids of given form, e. A more rigorous mathematical proof beside Hales' complete computer algorithm , for fcc packing of hard spheres as the most dense packing is still missing, and packing any other deformed spheres remains an open area of research.

One possible approximation is to expand the shape of a fullerene in terms of multipole deformations and study the 3D packing of such smooth topologies. Chemists can deduct many useful properties of a chemical system just by looking at its structure. For fullerenes, the distribution of the 12 pentagons on a surface, for example, a sphere, can tell us qualitatively how stable the fullerene is or how it would pack in the solid state.

Moreover, the symmetry of the underlying structure determines many useful spectroscopic properties. It is perhaps a realistic goal to connect the graph theoretical properties of a fullerene directly with its physical properties by mapping the fullerene graph G into a number describing that property. A topological index is called a chemical index if it is related to a chemical or physical property. Topological indicators may be placed in the same category as crude chemical bonding models, except that they can be strictly defined in graph theoretical terms and sometimes have interesting mathematical properties.

There is no restriction that the mapping between fullerene graphs and any topological indicator be one-to-one, and indeed, most of the commonly used topological indicators are the same for many different isomers. However, solving the electronic structure problem for any large fullerene is a daunting task and therefore, topological indices that are easily obtained can be very useful as we shall see.

Another example for a topological index is the 12 face spiral pentagon indices as described above. The first topological index of chemical relevance was introduced by Wiener in The Wiener index provided a good measure of compactness for acyclic alkanes and gave a reasonable correlation to boiling points. Many different topological indices have been introduced and studied since, mainly for the structural and statistical analysis of molecules, polyhedra, and graphs in general, often yielding interesting mathematical properties for a list of topological indices see for example Ref Some of these topological indices can directly be related to the stability of the fullerene cage.

We may contract the neighbor indices to one useful topological index describing the stability of fullerenes. For IPR fullerenes a more useful single topological index is defined through the hexagon signatures. We call the topological index H n the n-th moment hexagon signature.

For general fullerene isomers, low P 1 values and high H i values correlate with high thermodynamic stability. Fowler et al. The result is an efficient but effective screening method to find the most stable fullerene isomers. Ju et al. Note that for connecting two or three faces we have.

For example n is the sum of all occurring ring patterns containing the combination of two pentagons and one hexagon. The 15 basic shapes for the two- and three-ring face adjacencies on the surface of a fullerene; l denotes linear, b bent, o open, and c closed ring patterns see Ref This not only illustrates nicely the current development in the area of topological indices, it also suggests that the stability of fullerenes can be approximated by counting different face patterns. Indeed, Cioslowski et al. Alcami et al. For all fullerenes studied the rms error is The question is: How well are topological indices suited to describe fullerene stability?

This is indeed the case as the DFT results show. The different fullerene isomers are all very similar in surface area and volume. Moreover, these values are very sensitive to the method applied, and a better measure for the gap given by the singlet-triplet separation. We note that the two different fullerene isomers 3 and 5 have identical Wiener indices, and the isomer 6 with the smallest Wiener index is not the most stable isomer.

The symbol T for isomers 2 and 6 in the first column indicates that the triplet electronic state is taken instead of the singlet state. The first halma transform of C 20 is C 80 and therefore of I h symmetry. There are seven IPR isomers out of a total of 31, isomers as possible candidates for the energetically most stable one. With larger fullerenes the band gap becomes very small graphene is a semi-metal or zero-gap semiconductor , and one has to check for states of higher spin multiplicity.

The results show that the first three isomers are very close in energy and it requires perhaps a more sophisticated electron correlation treatment to sort out the sequence in stability. The symbol T indicates that the triplet electronic state is taken instead of the singlet state. The results shown here clearly demonstrate that the topological indicators help enormously to sort out the most stable isomers.

The situation is often quite complicated as seen from the DFT calculations here, or for example from the work of other authors who compared stabilities within a list of isomers. In graph theory, a perfect matching is a selection of edges such that every vertex of the graph G is part of exactly one edge in the matching. The edges of the matching correspond to the double bonds. Schmalz et al. Minimum, median, and maximum perfect matching count for all isomers of C N up to C The other two series behave similarly.

Narita et al. Figure 18 shows an AFM picture of one of the symmetry equivalent hexagons in C 60 clearly showing the two different types of bonds. These are the Fries and Clar structures. It is clear that a Clar structure localizes benzenoid structures in fullerenes. One might naively assume that a set of Clar structures form a subset of Fries structures, but this is generally not the case for fullerenes.

Double bonds are shown in red and isolated aromatic hexagons are shaded in. We expect maximum stability for fullerenes with the highest Clar number. For example, C 60 - I h is unique among all other isomers in that it has a Fries structure where all hexagons contain three double bonds and all pentagons none. This is also seen as a reason for the unique stability of C However, finding the Clar number is not a trivial problem as it is computationally NP-hard.

Hence the Clar number alone is not a good measure for stability. Figure 17 c shows that the expected number of perfect matchings is no different for IPR fullerenes than it is for all fullerenes, except for small fullerenes for which it is larger than average. However, the maximum number is always significantly lower for IPR fullerenes. There are, however, some good lower and upper bounds of these topological indices known.

This happens exactly for Clar structures that use every vertex. This gives an easy way to test whether a particular isomer admits a perfect Clar structure: simply test whether the inverse leapfrog operation is successful. Such fullerenes are called extremal fullerenes. Klein et al. For naming an alkene, the IUPAC rules state that one has to find the longest continuing carbon chain containing as many double bonds as possible.

For a cyclic system one chooses the longest cyclic chain, and if there are multiple longest cycles, one must choose the one that maximizes the number of double bonds along the chain. This is related to finding a Hamilton cycle , which is a closed path in a graph that visits every vertex exactly once. If at least one such cycle exists, we say that the graph is Hamiltonian.

In this case, the Hamilton cycles are the longest carbon chains, and the optimal one must be chosen among these. It is an open problem whether every fullerene has a Hamilton cycle. Because of this, naming fullerenes according to the IUPAC alkene rules is a computationally heavy task: we must search through all the exponentially many Hamiltonian cycles main rings to find the one in which the secondary bridges are labeled in the lexicographically smallest way.

In addition to being infeasible to calculate, as N grows, the name resulting for a fullerene from the IUPAC alkene rules rapidly becomes long and unmanageable.

## Edward Richtofen

Even for a relatively small fullerene such as C 60 - I h , the name k is already unreadable, and the name length keeps growing with the fullerene size. While computing the name of a fullerene according to this scheme is easy and efficient, and contrary to the alkene naming scheme is short and easy to understand, we find that this scheme is not much better than the IUPAC alkene nomenclature.

On the other hand, it does not uniquely specify a fullerene, since many isomers share the same vertex count and symmetry. We therefore advocate the use of one of the following two names to uniquely specify fullerenes, based on the generalized canonical face spiral pentagon indices FSPI , see the chapter on face spirals above.

Both schemes uniquely and compactly identify a specific fullerene molecule. For moderate values of N , we can specify the fullerene by its canonical index, which is the lexicographic number of its FSPI among the isomers. For the vast majority of fullerenes, this is a list of 12 small integers. For the extremely rare cases where jumps are necessary there are only two non-spiral fullerenes out of the 2. No currently known case requires more than one jump. In the latter example, the two numbers before the semicolon denote a cyclic shift of length 2 before adding face number The canonical FSPI notation is especially advantageous for more reasons than being compact and complete without needing to refer to a precomputed database.

Thus, one can even reconstruct fullerenes of moderate sizes from the FSPI by hand, without the help of a computer. The algorithm for the windup operation is , and the inverse operation, unwind , takes expected time to find a single generalized spiral, and to find the canonical one. Hence one can check whether two fullerene graphs are isomorphic simply by testing whether they have the same canonical FSPI, and the canonical FSPI gives us a unique graph representative of each isomorphism class via the windup procedure.

As discussed later, the FSPI representation makes it easy to directly compute the ideal symmetry group of the fullerene. Because the generalized face spiral algorithm is complete for all connected planar cubic graphs, a compressed form of the face spiral similar to the FSPI can be used in general for fulleroids, to be introduced in the last chapter. The spherical shape of C 60 - I h with no adjacent pentagons is seen as the main reason for its unusual stability, which underlines the importance of Kroto's isolated pentagon rule IPR.

Alternatively, for nanotube fullerenes with pentagon caps, the infinite particle limit is just the corresponding infinite nanotube. It was already shown experimentally that C 70 is more electronically stable than C It has been detected in by Prinzbach 16 see also Ref , and a current review on the state of affair concerning C 20 is given by Fei et al.

The graphene limit is estimated from the heat of formation of C It becomes more and more difficult to extract the most stable isomers from the huge isomer space as the size of the fullerene increases. Hence there should be a correction accounting for the curvature at each carbon atom. Clearly, as the fullerene system becomes larger, the curvature term becomes smaller. The smallest eigenvalue highest unoccupied level is largest for C 20 with , and for the IPR isomers C 60 we have.

As a geometric consequence, the famous golden ratio appears here as it does in the volume and surface area calculations discussed above. The computation of these coefficients soon becomes computationally intractable. The five or six Pauling bond orders in a pentagon or hexagon respectively can be added to give the Pauling ring bond order. We finally mention that fullerene cage abundance is not only guided by thermodynamics, but mostly by kinetic stability.

For example, the relative isomer abundance of fullerenes and carbon nanotubes correlates well with kinetic stability. However, as we have already seen, more detailed quantum chemical calculations are required to describe the bonding in fullerenes accurately see also the work by Thiel and co-workers — Moreover, the system becomes more multi-reference in nature, making even a quantum theoretical treatment difficult.

As an example, C 50 - D 5 h was investigated by Lu et al. In addition, some fullerenes may undergo first- or second-order Jahn-Teller JT distortions which could, however, be very small and almost undetectable for larger fullerenes , that is, they can distort to subgroups of the ideal point group symmetry given by the fullerene topology. It is therefore often difficult to predict the correct electronic ground state and corresponding physical point group symmetry of a fullerene. Since the discovery of fullerenes almost 30 years ago, there has been considerable activity, both from the experimental and theoretical side, to gain a detailed understanding of fullerene formation in the gas phase.

However, the formation mechanism and especially the high yield of C 60 - I h and C 70 - D 5 h remains elusive and somewhat controversial. Fullerenes can be produced by a evaporating a carbon target graphite, amorphous carbon, fullerenes , optionally with addition of metal oxides with a laser, 5 b an electric arc between carbon electrodes, 9 or c by partial combustion of carbon rich organic compounds.

Furthermore, fullerenes are found in space, 14 at meteor impact sites, after lightnings and bush-fires, and soot from household candles. The distribution of yielded fullerene cage sizes depends on the production method and the above-mentioned experimental parameters; however, C 60 - I h and C 70 - D 5 h are always among the most abundant species. A large number of formation mechanisms have been proposed. This mechanism can be divided into five phases: First, linear polyyne chains and cycles form.

In the second stage nucleation , entangled carbon chains rehybridize and form faces. It is to be noted, that pentagons and hexagons are close in energy at the given temperatures, 99 however, smaller and larger faces are formed as well. Third, more carbon dimers attach to the side chains of an existing nucleus growth , allowing for the formation of additional faces. The final step is the ejection of carbon dimers off side chains and the cage in combination with rapid isomerization of the cage structure resulting in a fullerene without side chains and faces of sizes 5 and 6 only. Fullerenes can not only shrink but also grow in steps of C 2 , 95 , 98 , and the existence of C 2 and C 3 fragments is backed by spectroscopy.

While the non-cage carbon concentration is high addition prevails; as the carbon vapor expands ejection begins to dominate. As shown in Figure 1 , large fullerenes are more electronically stable than small ones, with graphene having a lower energy than any fullerene. In an equilibrium one would therefore anticipate the formation of graphene—the formation of strained cages and especially the high yield of C 60 - I h and C 70 - D 5 h require further explanation.

The experimental conditions of the cooling and expanding carbon vapor are, however, far from an equilibrium, and the whole formation must be understood as a process of self-organization that is governed by kinetics more so than thermodynamics. Once fullerenes have formed they are subject to restricted equilibration only.

Curl et al. The experimental observation of endohedral metallofullerenes supports a top-down formation mechanism in two ways: As any enclosed fragments need to enter the cage before it is closed, fullerenes containing fragments that use up most of the space available in the carbon cage must either be formed top down or their existence implies the breaking and reformation of carbon bonds to open and close the cage after it was formed. Secondly, enclosed metal fragments may stabilize otherwise unstable fullerenes: Zhang et al. We consider only topological aspects here as there are several reviews on the chemistry of endohedral fullerenes available.

It is clear that the cavity in the fullerene cage should be large enough to encapsulate atoms, molecules, or even smaller fullerenes. Conversely to the minimum covering sphere problem MCS , we wish to find the largest sphere that is fully contained within the polyhedron. This is called the maximal inscribed sphere MIS , or simply inscribed sphere , and its radius for a given polyhedron P is.

In the case of convex polyhedra, the MIS is unique, and can be computed directly by finding the Chebyshev centre of the polyhedron. However, in the non-convex case, the MIS is no longer unique: Consider two overlapping circles in 2D space with the points lying on these circles.

We now have two equivalent possibilities for placing our inner circle. The same argument holds for peanut-shaped fulleroids, which are introduced in the last chapter. Hence, in the non-convex case, we only search for one of the possibly many largest spheres contained entirely inside the polyhedron, giving us a numerical optimization problem.

Once the MIS has been obtained, we can estimate whether an atom or molecule fits inside the fullerene without coming close to the repulsive wall of the fullerene cage. If we compare this to the Van der Waals radii for the rare gas atoms, which are 1. Even molecules like water fit into C In other words, the endohedral system can become repulsive very soon if the size of the endohedral atom or molecule becomes too large. As a further example we consider hyperfullerenes also called buckyonions , , , , that is, fullerenes which contain smaller fullerenes inside their cage.

They have already been detected and studied by theoretical methods. This only holds for ideal spherical systems, but can be seen as a lower limit for the guest C M fullerene. In fact, analyzing C 20 C 60 we find that the carbon atoms of C 20 close bonds with the carbon of the C 60 cage, that is, the structure should be considered as a carbon cluster rather than a hyperfullerene.

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In contrast, C 60 fits nicely into C , and C just into C in agreement with the analysis of Bates and Scuseria see Figure The buckyonions, which have been considered in the past, are all Goldberg-Coxeter transforms of C 20 and have the most spherical appearance as they are of icosahedral symmetry. It has been suggested that the rather high sphericity observed for buckyonions in experiments in contrast to the faceted polyhedral structure predicted see Figure 22 is due to C 2 removal in a [c] fragment consisting of 1 pentagon and 3 hexagons notation of Figure 16 is used here introducing one heptagon and an additional pentagon, that is, [l].

This gives a very simple estimate from Eq 49 for such fullerenes,. Finally, what happens if we relax the rules a little bit, and allow for other types of three-valent sp 2 carbon frameworks? There are many generalizations that lead to structures of beautiful shapes that have both elegant mathematical theory and physical realizations: allowing for polygons with faces different from pentagons and hexagons; for surfaces other than a sphere genus 0 , such as a torus genus 1 , a Klein bottle or a double torus genus 2 ; negatively curved Mackay-Terrones or Vanderbilt-Tersoff surfaces; etc.

We will discuss mainly two types of carbon frameworks: fulleroids, which are fullerene-like structures, and Schwarzites, which are periodic negative-curvature surfaces. Different authors employ a wide range of working definitions for fulleroids. We advocate the following definition, generalizing fullerenes: A fulleroid graph is a three-connected trivalent polyhedral graph.

The definition is identical to that of fullerenes, except that there is no restriction on face sizes. Hence, a fullerene is a fulleroid with only pentagon and hexagon faces. We further generalize to genus-n fulleroids , abbreviated G n -fulleroids, defined as the three-connected trivalent genus- n graphs. What kind of fulleroids are allowed? Can we tile a sphere or a torus with heptagons only the answer is no , or with only pentagons and heptagons?

And how can we construct such fulleroids? Consider a possibly irregular genus- n graph with N vertices. Let N r be the number of vertices with degree r , and F n be the number of n -gonal faces. Then we trivially have. If we only allow for three-valent graphs fulleroids , the first term on the left hand side of Eq 53 conveniently vanishes hence their special place in graph theory. This gives, From this equation we make a number of observations. The hand-shaking lemma last formula in Eq 52 allows only for an even number of vertices. We already know that there exist no polyhedral structures of the type C 22 [5,6].

To give another example, for C N [5,7] polyhedra with pentagons and heptagons only no hexagons one requires at least two heptagons since, analogous to C 22 , the combination of 13 pentagons and one heptagon is not valid despite being allowed by Euler's formula. Because of this, the smallest [5,7]-fulleroid is C 28 - D 7 d with 14 pentagons and 2 heptagons, as shown in Figure These can be obtained by replacing a [c] fragment by [l], and can even be energetically favorable.

For example, Fowler and co-workers showed that C 62 consisting of one heptagon, 13 pentagon, and 19 hexagon faces is of lower energy than all the regular fullerene isomers of C The torus is the only closed, orientable surface that can be tiled exclusively with hexagons. However, the plane, which is not closed, can be tiled with hexagons as well graphene , as can the Klein-bottle which is not orientable. Tiling in a pattern with alternating pentagons and heptagons yields a toroidal arrangement of azulenes, called azuloids.

A few examples should be mentioned here. As a result, structures deviating from the classical fullerenes can be more stable energetically or lie close by. For example, for C 26 , An et al. Kirby and Pisanski showed how 2D graph drawings of toroids can be obtained. Faghani analyzed the symmetry of toroidal fulleroids. As discussed earlier, the positive curvature in fullerenes originate from the pentagons, because sheets of hexagons like to be planar. Introducing heptagons or even octagons into a fullerene requires additional faces of size less than 6 to outweigh them, and results in a highly non-convex structure with often strong negative local Gaussian curvature, introducing a saddle-type topology to the structure.

Two examples of such fulleroids are shown in Figure In Figure 26 , heptagons are introduced in a fashion that leads to peanut shaped fulleroids, and in fact, many types of interesting shapes can be constructed. Spiky fulleroids with negative curvature containing heptagons derived from fullerenes through patch replacement.

Peanut shaped fulleroids. More interesting fulleroid shapes can be found in a recent paper by Diudea et al. Structures of negatively curved graphitic carbon, which can be periodically extended to a lattice, have been proposed in by Mckay and Terrones, and subsequently explored by experiment. The two structures in Figure 27 are termed P-type and D-type, and can be approximately represented by the simple formulae derived from the Weierstrass-Enneper parameterization for minimal surfaces, Relevant reviews on such minimal surface carbon networks have been given by Terrones and Mackay and by Terrones and Terrones.

Lenosky et al. In principle, Schwarzites can be associated to any kind of lattice, either periodic or amorphous, the latter are realized in so-called random Schwarzites observed experimentally. A recent comprehensive review over experimental and theoretical studies on Schwarzites has been given by Benedek et al. Comparison between a small portion of a TEM image of a P-type Schwarzite and the surface described by Eq 55 projected onto a plane normal to the z -direction see Ref for details. Finally, we consider non-cubic n -valent polyhedral structures. In chemistry this is realized by using elements from the periodic table which can share more or less than three bonds with its neighbors.

For example, Au 32 and Au 72 are a triangulated surfaces of icosahedral symmetry. We may ask how many possible isomers are there for Au 72?

Since the mapping from the fullerene isomer space into the dual space is bijective, there are as many isomers for Au 72 as there are for C , and there are 7,, of these. The stability of these structures have not been explored yet! A smaller dual-type golden fullerene structure, Au , has been observed spectroscopically by Bulusu et al. Only the Au - T d structure has been investigated presently. All boron fullerene B 40 adapted from Ref As a second example, the all-boron B 40 fullerene has been identified by photoelectron spectroscopy only very recently by Wang and co-workers as the anionic species B , consisting of 4 heptagons, 2 hexagons, and 48 triangles see Figure In this intriguing new polyhedral structure the vertices are of degrees 4 and 5.

To review the many developments in the topology and graph theory of fullerenes or fulleroids would be a monumental task. We have only outlined a few important concepts in order to give the reader a good introduction into this exciting field. There is certainly the need of a more comprehensive review or book. Nevertheless, we hope to have shown that the interplay between mathematics and structural chemistry is both interesting, rich and well alive.

As more sophisticated method in the synthesis of carbon and other materials become available, we hope that some of the exotic, but very beautiful, structures become accessible with many useful applications in chemistry and materials science. PS is indebted to the Alexander von Humboldt Foundation Bonn for financial support in terms of a Humboldt Research Award, and to both Gernot Frenking and Ralf Tonner Marburg for support during his extended stay in Marburg where the topology project began.

For definitions see below. Consequently, every planar embedding of such a graph defines the same set of faces. The converse it true as well: only three-connected graphs have this property. In the discrete case they are the same, except that Gauss-Bonnet is stated in terms of the Gaussian curvature. In practice, however, it turns out to not matter. P is the complexity class of counting solutions for decision problems in NP.

While only exponential algorithms exist both for NP-complete and P-complete problems, in practice P-complete problems are dramatically harder than NP-complete. However, it is a long standing but yet unproven conjecture that fullerene graphs, a small subset of the cubic planar graphs, are all Hamiltonian. It has been verified by Brinkmann, Goedgebeur, and McKay for all 2. National Center for Biotechnology Information , U. Wiley Interdisciplinary Reviews. Computational Molecular Science. Published online Oct Author information Copyright and License information Disclaimer.

This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. This article has been cited by other articles in PMC. Abstract Fullerenes are carbon molecules that form polyhedral cages. Open in a separate window.

Figure 1. Figure 2. Planar connected graphs fulfil Euler's polyhedron formula ,. Figure 3. Drawing Fullerene Graphs: Methods for Planar Embedding Planar embeddings can be drawn on paper and give us a way to visualize the structures of planar graphs. Figure 4. Figure 5. Face Spiral Representations of Fullerene Graphs One of the first methods for encoding fullerene graphs was the face spiral algorithm by Manolopoulos et al. Figure 6. Figure 7. Generation of Fullerene Graphs In order to explore chemical, physical, or graph theoretical properties for a wide range of fullerenes, it is important to have access to a list of stored fullerene graphs.

Transformation of Fullerene Graphs Transformations of fullerene graphs can be divided into local transformations that leave all but a certain region of the graph unchanged, and global transformations such as the Goldberg-Coxeter transformation. Figure 8. Figure 9. Figure Geometry of Fullerenes Many of the beautiful properties of fullerenes derive from their relation to algebraic and differential geometry.

Generating Initial Structures Before optimizing a fullerene structure by for example a force-field method, we need a reasonable initial structure. Fullerene Force-Field This section describes how to obtain good results for the molecular geometry numerically by way of specially tailored force-field optimization methods. Fullerene Symmetry There are two symmetry groups associated with a fullerene: the ideal or topological symmetry group of the fullerene graph, and the real or physical symmetry group of the molecule in 3D space.

Topological and Chemical Indicators Chemists can deduct many useful properties of a chemical system just by looking at its structure. Thermodynamic Stability and the Graphene Limit The spherical shape of C 60 - I h with no adjacent pentagons is seen as the main reason for its unusual stability, which underlines the importance of Kroto's isolated pentagon rule IPR. The Gas Phase Formation of Fullerenes Since the discovery of fullerenes almost 30 years ago, there has been considerable activity, both from the experimental and theoretical side, to gain a detailed understanding of fullerene formation in the gas phase.

Endohedral Fullerenes and Buckyonions We consider only topological aspects here as there are several reviews on the chemistry of endohedral fullerenes available. The buckyonion structure C 60 C C Weird Fulleroidal Shapes: Generalizing Fullerene Structures Finally, what happens if we relax the rules a little bit, and allow for other types of three-valent sp 2 carbon frameworks? Acknowledgments PS is indebted to the Alexander von Humboldt Foundation Bonn for financial support in terms of a Humboldt Research Award, and to both Gernot Frenking and Ralf Tonner Marburg for support during his extended stay in Marburg where the topology project began.

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