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Let's go back 10,000-12,000 years, when humans have yet to make strides in basic machines. In our world, the invention of the wheel makes things easy to move, starting a transportation revolution. This will eventually lead to new technologies, and eventually the rise of more complicated machines.

What if the wheel was never invented? In this world, humans discover a gel-like substance that, when put on an object, immediately reduces friction. All of a sudden, pushing a boulder across a grassy field is more like pushing a boulder across an icy pond - but even easier. The effects of friction can be almost totally neglected.

This leads to a different type of transportation revolution. All of a sudden, the coolest new way to get around is in large sleds with their undersides coated with this gel. Soon, horses are domesticated - way ahead of schedule - as it becomes clear that putting them side by side in front of a sled is a great way to pull heavy loads.

Will this society ever be motivated to invent the wheel (assuming that pulleys are created not out of axles, but out of gel-coated blocks)?

Related points can be found in In a society of flying beings, would the wheel ever be invented?, especially related to pulleys and the application of the wheel to transportation.

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    $\begingroup$ It seems extremely unlikely that such an elementary tool would never be invented, even by a civilization that reached the technological age we now find ourselves in. Depending on the properties of your magic gel it might never have a practical application, but given the mathematically interesting properties of circles, I would be surprised if engineers could avoid ever creating one. Gel scarcity or shortage is also a real concern to motivate wheel invention / adoption. $\endgroup$
    – Avernium
    Commented Dec 21, 2015 at 22:13
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    $\begingroup$ Not to mention all the slimy gel trails left everywhere. Eventually there would be a lot of buildup. People would be tracking it into the house, and slipping and breaking things. Bleh :) $\endgroup$
    – Francine DeGrood Taylor
    Commented Dec 22, 2015 at 1:16
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    $\begingroup$ @FrancineDeGroodTaylor Hopefully it washes away eventually! Otherwise, I anticipate a huge market for cleaning supplies. . . $\endgroup$
    – HDE 226868 ♦
    Commented Dec 22, 2015 at 1:17
  • $\begingroup$ What about highly uneven ground? Stone and small ledges would stop a sled in a way that wheels may be able to roll over with enough force. A sled may need to be lifted to get over things that wheels could handle regardless of friction. $\endgroup$
    – shiningcartoonist
    Commented Dec 22, 2015 at 15:31
  • $\begingroup$ @shiningcartoonist I think it depends on the sled you're thinking of. If it has a curved front (it's not a wheel!), the lack of friction might get you over ledges that a wheel would roll over. $\endgroup$
    – DoubleDouble
    Commented Dec 22, 2015 at 16:31

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Historically, there have been many societies that never invented the wheel. We really only got rolling with wheels in the bronze age, and not everyone had them. For example, there's no evidence that the natives of Central America ever used wheels for anything beyond small children's toys. Where they were invented, they actually came after "more advanced" technologies like sailing ships. So your premise is sound: the wheel is not a primary invention.

However, I find it unlikely that a society could become industrialized without adopting the wheel. It has many uses where low-friction linear motion just won't do: flywheels, electric motors and generators, millstones(where you need high friction), gearing in all sorts of mechanical things (clocks being the obvious example, but pretty much any technology you'd associate with steam-era industrial society relies on the existence of a wheel in one way or another), etc etc.

Having a substance to dramatically reduce friction in a robust variety of circumstances like you describe would certainly change a lot about how technology evolves in that culture, but that substance can't replace wheels in every instance where we use them now. And even if it could, how abundant is it? The economics of "gel mining" might push someone to start using wheels for transportation as a more cost-efficient alternative.

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    $\begingroup$ One point about Central America: The societies there didn't have access to animals like horses or mules, that could otherwise be used to pull things. Do you think that things might have been different otherwise? $\endgroup$
    – HDE 226868 ♦
    Commented Dec 21, 2015 at 22:05
  • $\begingroup$ @HDE226868 the wheel is extremely useful in transporting heavy goods also without draft animals - see lowtechmagazine.com/2011/12/the-chinese-wheelbarrow.html for examples. $\endgroup$
    – Peteris
    Commented Dec 22, 2015 at 7:44
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    $\begingroup$ I would say the lack of draft animals was definitely a contributing factor, and possibly the largest factor. Other things may have also played a role (the difficulty of technological diffusion on the N-S axis of a continent, for example). Jared Diamond's book, Guns, Germs and Steel, discusses the reasons for this sort of thing in a lot of depth. I highly recommend it, it will help make your worlds more realistic (and it is informative about the real world). $\endgroup$
    – realityChemist
    Commented Dec 22, 2015 at 14:16
  • $\begingroup$ Do you think they could get by with other curves of constant width? $\endgroup$
    – Christopher King
    Commented Sep 23, 2016 at 19:08
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    $\begingroup$ Poul Anderson used constant width curves for rollers in an sf story where circles were forbidden for religous reasons. (The Three Cornered Wheel, included in The Van Rijn Method from Baen Books) That avoids the issue with axles, but replaces it with a mechanism to catch and reuse the rollers coming out at the back.... $\endgroup$
    – JerryTheC
    Commented Sep 23, 2016 at 21:38
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The wheel will always be discovered given sufficient time and chance.

Nature does not abhor a wheel. Natural wheels exist in a variety of forms.

Dung beetles form their meal into a ball and roll it around. Trees form logs that tend to roll down hills on occasion. Eyes are balls that rotate in a sockets. Knees are obvious ball and socket joints.

Someone is going to forget to apply the sliding gel and when they attempt to push a log, the log will respond by rolling.

Sure, people are often blind to the obvious. But even the Aztecs had wheels, they simply did not see the need to apply them in their society. Had they become more industrialized, I have no doubt that they would have been using wheels in more than toys.

On soft soils and irregular terrains wheels are not ideal. Maybe that is partially the reason the Aztecs never made use of the wheel. But given their constructions, one wonders why they never invented the wheelbarrow. The Aztecs even made roads and bridges but not wagons. But the fact is that the wheelbarrow was apparently not known in Europe until about 1200 AD. This just seems incredible to me. Romans had chariot races in the Coliseum, but no wheelbarrows to help in its construction.

So just because something seems incredibly obvious in hindsight, it does not mean that it is actually obvious. Still, given enough time and potential observers looking for possible applications of a technology, someone will see the blindingly obvious for what it really is.

By the time you've added physicists and mathematicians to society, the use of a sphere is self-evident.

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Yes

The Aztec Empire lasted for around 1000 years and their largest city was larger then any European city at the time, and they never discovered the wheel. In fact, no New World culture invented the wheel before meeting an eastern power. We don't know why they didn't invent the wheel for themselves, but they didn't so it is possible. My suggestion for a culture without a wheel is simply to have their environment be too difficult for the use of wheels, like a jungle, swamp or mountain.

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    $\begingroup$ No, no, they had wheels, they just used them as children's toys. They had no domestic animals suitable for pulling them. Though there's no excuse for not inventing wheelbarrows, but, as noted above, Europeans invented wheelbarrows long after carts. $\endgroup$
    – RedSonja
    Commented Dec 22, 2015 at 8:24
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    $\begingroup$ Which I doubt, actually. The first blacksmith who saw a chariot must have built something similar to use in the garden, surely. Just because archeologists never found one doesn't mean they weren't in use. $\endgroup$
    – RedSonja
    Commented Dec 22, 2015 at 8:26
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    $\begingroup$ @RedSonja or maybe a carpenter, hence why nothing has been found. $\endgroup$
    – Chris H
    Commented Dec 22, 2015 at 12:13
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    $\begingroup$ @RedSonja, it also depends on how you define a wheelbarrow. Typically what we mean when we say wheelbarrow is a cart with a single wheel at the front and handles at the back (a class 2 lever), while a typical chariot has two wheels almost exactly under where the passenger stood (not really a lever at all). It seems obvious to us, but it may have taken quite a while for someone to make that change. $\endgroup$
    – realityChemist
    Commented Dec 22, 2015 at 14:23
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The wheel and rotational motion are almost impossible to separate. If you develop rotational motion, you'll almost certainly end up developing a wheel by accident (or a pulley or whatnot).

Now if there was a very easy way to minimize friction for transportation purposes, we may never decide to use the shape of a wheel for transportation, but we'll certainly invent the idea that circular things work better for rotation.

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The Inca had llamas that carried burdens,and sometimes pulled a narrow sled, but the roads were not accommodating for a wheeled cart.

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    $\begingroup$ can you expand a little more on your answer, maybe add a couple links about their roads? $\endgroup$
    – bowlturner
    Commented Dec 30, 2015 at 17:53

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