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My textbook says the following:

Unique among the elements, carbon can bond to itself to form extremely strong two-dimensional sheets, as it does in graphite, as well as buckyballs and nanotubes.

Is carbon the only element that can do this?

If not, then what are the other elements can also do this? Is there a term to describe such elements?

What is the chemical characteristic that allows this to occur?

I would greatly appreciate it if people could please take the time to clarify this.

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    $\begingroup$ Catenation : self linking property of an element (atom). $\endgroup$
    – glucose
    Apr 26 '19 at 10:47
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    $\begingroup$ The right question isn't whether an element can bond to itself. Plenty do that. The issue is whether an element can form a wide variety of stable structures when bonded to itself. $\endgroup$
    – matt_black
    Apr 26 '19 at 13:32
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    $\begingroup$ I was also aware that boron sheets were recently synthesized, but did not realize that someone has even predicted nitrogen sheets that could be stable at room temperature! $\endgroup$
    – jeffB
    Apr 26 '19 at 15:20
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    $\begingroup$ Sulfur can also form rings and chains $\endgroup$
    – porphyrin
    Apr 26 '19 at 15:52
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    $\begingroup$ What is the actual question here? Your citation does not say that bonding between identical atoms is something unique - every single element could do this even if transiently. The point is that carbon carbon allotropes are somewhat exceptional. $\endgroup$
    – Mithoron
    Apr 26 '19 at 21:22
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Is carbon the only element that can do this?

No, carbon is not the only element with such characteristics.

If not, then what are the other elements can also do this?

There is a whole number of elements such as silicon, arsenic, germanium.

Is there a term to describe such elements?

At least I'm unaware of such a term, which might be furnished by our far wiser community.

What is the chemical characteristic that allows this to occur?

Catenation.

More information:

According to the Molecular Orbital Theory, the condition for a compound to exist is that it should have more electrons in the bonding orbitals than in the anti-bonding orbitals. So, as long as you have the bonding orbitals filled more, you can have pretty anything, more than just chains of atoms.

Thus, the existence of a compound also depends on the precise conditions in which the compound is kept, for example sodium forms different types of chlorides under different conditions and that as pointed out by Poutnik in the comments, $\ce{He2^1+}$ and a ton of others are discovered and still more awaiting discovery.

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No, carbon is not the only one that can bond to itself. It's a unique property of some elements mainly the group 14 elements like silicon, germanium, arsenic etc. This phenomenon is called catenation. It might be mainly due to presence of four valence electrons in their outermost shell. A large number of carbon atoms are linked with each other with sigma and pi bonds. However catenation gets limited as we move down the group in group 14.

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You wanted to know if any elements were able to match carbon's ability to form 2D sheets, nanotubes, and buckyballs.

According to https://www.azonano.com/article.aspx?ArticleID=4863 and other sources, graphene-like 2D sheet formation has (thus far) been demonstrated in the following elements:

boron, silicon, germanium, phosphorus, antimony, bismuth, arsenic, and tin

In addition, nanotube formation has (thus far) been demonstrated for all of the above except tin.

Since any element that can form a 2D sheet should be able to form a nanotube, it is expected that tin nanotubes will be developed eventually. In the meantime, computational studies have been done to predict the properties of tin nanotubes.

By contrast, the subset of the above list that has been shown to form buckyballs is, thus far, limited to boron and silicon.

The formation of buckyballs (60 atoms) involves bond length and angle constraints, which may rule out some of the elements that can form 2D sheets. Though it is possible that elements that cannot form buckyballs could form fullerenes of other sizes.

I've repeatedly added the "thus far" qualifier to indicate that these properties may eventually be demonstrated for other elements as well.

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So most of this has been answered, but the main elements that can bond to themselves are called diatomic atoms. This includes hydrogen, nitrogen, fluorine, oxygen, iodine, chlorine, and bromine. There are other elements that can do this as well, however these main elements occur naturally in their diatomic state as gases.

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    $\begingroup$ "Diatomic atom" is a weird term, diatomic molecule makes way more sense. Besides, the way the answer is formulated, it appears to me that all elements that are capable of self-bonding are diatomic molecules, whereas in reality diatomic molecules is a tiny subset of the former. $\endgroup$
    – andselisk
    Apr 26 '19 at 20:30
  • $\begingroup$ It's pretty clear that the question is asking about self-bonding beyond simple diatomic molecules. $\endgroup$
    – Mark
    Apr 26 '19 at 22:24
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    $\begingroup$ The ability to make a stable diatomic molecule might be the reason for not observing chains or 2D structures for these elements. $\endgroup$ Apr 27 '19 at 3:56
  • $\begingroup$ @Karsten Theis very much agree. Extreme Peierl distortion. $\endgroup$
    – Alchimista
    Apr 27 '19 at 7:22

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