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An ideal sheet of paper, a plain, is a two dimensional object and has by definition no volume. In case of Gabriel's horn we really have a three dimensional obje
by danbruc 10y ago
An ideal sheet of paper, a plain, is a two dimensional object and has by definition no volume. In case of Gabriel's horn we really have a three dimensional object bounded by a two dimensional surface. The bounding surface itself has no volume just like a plain.
- ktta 10y agoBut an ideal sheet of paper can have finite surface area and infinite perimeter length. See the Stackoverflow link in a comment above.
- danbruc 10y agoOf course, in both cases it is interior - volume respectively area - and boundary - surface area respectively boundary length - but that is not the same thing as area and volume of a two dimensional object. The analog for Gabriel's horn would be volume and four dimensional hypervolume which is of course also zero because it is a three dimensional object. The paradox goes from interior to boundary, what mkagenius suggested is taking the interior one dimension up.
- ezequiel-garzon 10y agoI cannot resist: it's a plane, not a plain.
- danbruc 10y agoDamn it, I even looked it up because I was unsure. But the German word Ebene translates to plane and plain and I missed that.
- deleted 10y ago[deleted]