After the plots are surveyed and plotted on a map (I am assuming this was done at some point), even irregular, curvilinear shapes should not matter as you can measure out the areas using a planimeter or some other method.
On the other hand, if the plots are (nearly) rectangular, then you only need to measure 2 sides to calculate the area! (And for non-rectangular quadrilaterals that weird formula strictly overestimates the area anyway. I hope it was not introduced in order to rip people off.)
Our land plot is a convex pentagon so, although that bulge is only slight, the simplified measurement isn’t totally against us. (We measured from quadrilateral corner to corner, rather than following the bulging edge.) But now that you mention it, maybe it was a way to rip us off! (And the land has a slope, also leading to an exaggeration of its area.) All the witnesses seemed to think the area calculation was quite ordinary … but father-in-law was one of the few witnesses on our team. :eek:
At only about $750/acre, that land measurement will not make the Top Fifty List of times I’ve been defrauded. OTOH, the land did not come with a land deed.
The government did eventually survey the land and issue a document. I think the document reads “This acknowledges that you are squatting, and paying taxes, on degraded forest for agricultural purpose. This squatter’s permit may not be bought or sold. It may be hypothecated only at the Government Bank for Agriculture.” Every decade or so a political party wins election promising to issue land deeds. Their efforts are quickly squashed by the kleptocrats who derive advantage from rural poverty.
Despite the vast difference in their economic training, top World Bank economists and poor-country rural farmers agree on one thing: Getting land deeds for their land would be a a big help financially.
Texas Farmer - “On my ranch, I could get on my horse and ride all day and not reach the edge of my property.”
Kentucky Farmer - “I used to have a horse like that too…”
Learning about this and calculating the volume and surface area in high school Calc was when I realized I wouldn’t be taking any higher level mathematics.
There should be an excellent political jab possible with a name like “Koch snowflake”, but since this is GQ I will refrain.
I’ve never understood why Gabriel’s Horn was considered so bewildering or counterintuitive. There are lots of ways to get a shape with arbitrarily large or infinite surface area, but finite volume. For starters, just take any of the two-dimensional figures with an arbitrarily large perimeter-to-area ratio, and extrude it some finite amount.
I’m not sure which “two-dimensional figures with an arbitrarily large (infinite?) perimeter-to-area ratio” you’re thinking of, but Gabriel’s Horn is not a fractal. Nowadays, mathematicians are used to believing six impossible things before breakfast, but Gabriel’s Horn was devised at the dawn of calculus, when calculating such volumes and areas first became easy. This was before the time of Newton and Leibniz; the generation of divergent series was viewed with suspicion; and [from Wikipedia] “Torricelli tried several alternative proofs, attempting to prove that [Gabriel’s Horn’s] surface area was also finite - all of which failed.”
And the procedure for painting the Horn must seem a bit peculiar, if not “paradoxical.” You can’t paint its infinite interior surface with a finite amount of paint, right? But you can fill the Horn with a finite amount of paint so, if the paint is adhesive, you’re done!
You can paint it easily, and you do not even need that much paint. Just keep the paint layers infinitely thin.
The fractals are easier to tile and stack up in a cabinet, though.
Sure, Gabriel’s horn isn’t a fractal. But on the other hand, it does have an infinite length, and it’s not much surprise that an object with infinite length would have some weird properties.
Here’s another object with a finite volume and infinite area: Take a circle, and then draw a ray off of it, and now extrude it. You’ll get a cylinder with an infinite plane segment attached to it. All of the volume is in the cylinder part, which is finite, but the plane segment contributes an infinite surface area. And it’s also completely boring.
It only takes an infinite amount of paint to paint the horn, meanwhile, if your paint layer is of constant thickness. Get too far down it, and that thickness won’t fit inside the interior.