Flatland, an 1880 novella by Edwin A. Abbott, is about a world with just two dimensions, inhabited by lines and polygons. Trouble starts when a sphere shows up.

We have the same kind of trouble today. No matter how useful Linux and the Internet prove to be, most people have trouble getting their heads around them. That's because most of us are built to see the world's organizations — especially those of business and government — as two-dimentional hierarchies. But Linux and the Net aren't hierarchical. They are heterarchical. Or are at least meant to be.

The Net was designed at its start as a system of autonomous and self-empowered end points that are all at zero functional distance from each other. This is the "distributed" (opposed to centralized) network model Paul Baran first described in the early 1960s. This model shaped the ARPANET which became the Internet. Baran drew his distributed network as a field of dots with lines between them. That was the best he could do using only two dimensions. But he really needed three.

How could one best draw a network in which any node could connect with any other node? Craig Burton came up with the best answer, which I wrote up in an column here twelve years ago titled The Connectivity Infrastructure: "a hollow sphere in which every point is visible to every other point across an empty space in the middle — a vacuum where the virtual distances are zero". He also added that putting a controlling intelligence in the middle would subtract value from its zero-ness.

It's hard or impossible to see that sphere if your head stays inside the flat triangle we call a hierarchy. Even complex hierarchies, such as those of giant corporations, governments and military organizations, are always depicted as two-dimensional and roughly triangular org charts, flat as a whiteboard.

The word hierarchy is older than English. Born as the Greek hierarchēs ("leader of sacred rites"), it evolved into Latin (hierarchia) and then through French into the Middle English jerarchie before arriving at its present English spelling in the 14th century. The current (February 7, 2014) edit of Wikipedia's hierarchy article adds this technical jive to the dictionary definition:

Abstractly, a hierarchy can be modeled mathematically as a rooted tree: the root of the tree forms the top level, and the children of a given vertex are at the same level, below their common parent. However, a rooted tree does not allow for items to be "at the same level as" one another, since a tree prohibits cycles. To accommodate this, a hierarchy can be modeled using a graph or a pre-order relation on the set of items. Alternatively, items of like type can be grouped together, and the hierarchy can be modeled using a partial order relation on the set of sets-of-like-items.

Got root? If so, you're at the top of a hierarchy like the one described in that paragraph, and possess power that fans down through directory paths. But your power is over your own system—one node on the surface of that giant sphere.

The Linux kernel itself is produced by a hierarchy that Kernel.org describes as "Linus Torvalds with assistance from a loosely-knit team of hackers across the Net". At the top is Linus, above a few dozen maintainers. Below that are any number of patch submitters. Yet the purpose of that hierarchy in kernel space is to support the absence of hierarchy in userspace. Unlike operating systems from Apple, Microsoft and Google (which bases Android on Linux), Linux serves no corporate agenda, meaning it belongs to no hierarchy. The same goes for the Internet in which Linux was born and throughout which it continues to grow and evolve.

Even though it supports an infinite variety of hierarchies, the Internet is not hierarchical. Instead, like Linux, it belongs to a far less talked about class of organizational being: heterarchy. Derived from the Greek heteros (other, different) and arche (sovereignty), heterarchy has relatively few examples, none of which make full sense in hierarchical terms.

Adriana Lukas explains, "Heterarchy poses an alternative to hierarchy itself, rather than another evolutionary stage of hierarchy. One comes from scarcity and the other from abundance." Business and government are both built in hierarchical forms to manage scarcities. Meanwhile, virtual worlds created by digital technologies and the Internet's protocols are abundant beyond full reckoning, and as alien to scarcity-based mentalities as a sphere to a polygon.

To help get our heads around this sphere, here are Adriana's "Five laws of heterarchy", with brief explanations of each:

1. Collapse of functions at the node level. "For a heterarchy to exist and persist, each node has to be able to perform certain base functions, meaning no hard-wired distinctions or divisions of role and functionality among the nodes."
2. Freedom and ability to bypass, that is, to choose a different path and thus avoid obstacles, control and imposition of hierarchy via backdoor. "An example of this is John Gillmore's familiar statement, 'The Net interprets censorship as damage and routes around it'".
3. Decentralized and distributable resources. "This is the difference between having to go to the post office to make a phone call and the ability to make a call at home or on your mobile phone. This law is important because having to use a centralized resource, apart from restricting nodes' autonomy, opens up opportunities for control of that resource and imposition of a hierarchy."
4. Abundance of resources or at least abundance of the most important resource that enables to create and maintain the network. "Don't try to build a heterarchy around a scarce resource, because once that resource is controlled, heterarchy disintegrates. An example of an abundant resource is information online. Digital format makes information duplicable and therefore abundant."
5. The marginal cost of communication needs to be zero or near zero. "This is important because in a peer-to-peer network it takes many more information exchanges to negotiate transactions."

She also says "asymmetrical balance appears to be a general feature of peer-to-peer/heterarchical networks", and provides two examples to explain what she means:

TCP/IP turns any server into an originator, relaying party or recipient of a message. The relationships between nodes are not symmetrical, but each server at any time can perform any role. Hence the term asymmetrical balance.

BitTorrent is another example—once a user starts downloading a file, he or she is automatically uploading it too, that's the inbuilt balance. But the downloads and uploads are not reciprocal: a different set of nodes are providing files for me to download and different set of nodes are downloading from me, so it's asymmetrical.

Adriana credits World of Ends with influencing some of her thinking on heterarchy, which is still in a formative stage. World of Ends in turn was influenced by thesis #7 of The Cluetrain Manifesto: "Hyperlinks subvert hierarchy", which was coined by David Weinberger. Hyperlinks, Adriana says, are heterarchical, exemplifying all five of her laws. As a kind of corollary to David's thesis, Adriana adds, "Networks dissolve hierarchy."

I suspect that the laws of both hierarchy and heterarchy apply throughout nature, and often dissolve each other. We see this happening live in higher education today. On the hierarchical side, universities value the abundance of knowledge in the world, yet organize knowledge sharing within hierarchical systems that are thick with command, control and costs. On the heterarchical side, universities are having their hierarchical gears stripped by the ability of pretty much anybody to learn pretty much anything by connecting to published knowledge (and other people), across the vast heterarchy of the Internet. (The challenges and opportunities here are expressed in the title of David Weinberger's latest book, Too Big to Know: Rethinking Knowledge Now That the Facts Aren't the Facts, Experts Are Everywhere, and the Smartest Person in the Room Is the Room.) The room is a heterarchy.

I also believe freedom thrives in heterarchy. Consider the freedoms listed in The Free Software Definition and how well they align with Adriana's five:

• The freedom to run the program, for any purpose (freedom 0).
• The freedom to study how the program works, and change it so it does your computing as you wish (freedom 1). Access to the source code is a precondition for this.
• The freedom to redistribute copies so you can help your neighbor (freedom 2).
• The freedom to distribute copies of your modified versions to others (freedom 3). By doing this you can give the whole community a chance to benefit from your changes. Access to the source code is a precondition for this.

These are embodied in the General Public License (GPL), which Linux adopted in the beginning (with Version 2). That was 24 years ago. Today business still has trouble grokking free software and the GPL, because nothing in either suggests ways to manage scarcity. From the standpoint of hierarchy, the fact that Linux has floated $trillions in economic activity is irrelevant. When positive externalities are external to the model framing one's view, they remain out of sight.

The same is true of the Internet. The difference is that nearly all of us access the Internet through ISPs that make it scarce to some degree. Thus, by controlling pathways in the network (and violating Adriana's fourth law), ISPs dissolve some of the Internet's virtues as a heterarchy, and with it the capacity of the Internet to float $trillions in economic activity, rather than mere $billions for ISPs alone.

It helps to recall that the Internet was not born as a "telecommunications service" or an "information service" (the US legal classifications for telephony and cable) or even a "service" at all. It is a state of connectedness made possible by protocols that were not created for billing purposes. Instead they were created—whether their authors knew it or not—in compliance with the laws of heterarchy. Those are the laws that create space for what Bob Frankston calls "DIY connectivity".

It is interesting that ways have been found to make Linux scarce to some degree, through cloud service providers, such as Amazon and Rackspace. One appeal of these services is that, as Jamie Zawinski once put it, "Linux is only free if your time has no value". Time is scarce for all of us, even when we operate in heterarchies. Yet operating Linux in clouds has become cheaper and cheaper. It is now reasonable to assume that, at some point, the marginal cost of cloud computing and storage will get so close to zero that the fifth law of heterarchy will apply, along with the other four.

And maybe that will happen with the Internet as well, in due time. Once the economic benefits of cheap access to the free and open Internet become fully obvious, business will rush toward the same equilibrium between hierarchy and heterarchy.

But we'll reach that stage a lot faster if we get our collective heads out of the triangles in Flatland and wrapped around the vast spheres of achievement and opportunity floating in our midst. Understanding heterarchy should help with that.