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Facts

Paradoxical perimeter

By 3. March 2019November 27th, 2020No Comments

Facts

Paradoxical perimeter

2019-03-03

Imagine the earth would be a perfect sphere and around this sphere runs a band at the equator, directly on the sphere.

The band is now cut at one point and a piece of band 3 m long is inserted. In a miraculous way, the band rises evenly over the equator and floats perfectly above it.

Question

How large could the distance between the band and the earth be? Does a finger, hand or arm fit between the band and the earth? Do we have to worry about the safety distances according to ISO 13857? ISO 13857 specifies that for such a slit-shaped opening:

  • for fingers not more than 6 mm,
  • for hands not more than 30 mm and
  • for arms no longer 120 mm.

And how does the distance with such a 3 m supplement behave with diameters other than the earth diameter?

In comparison to the total length of the band that lies around the earth, 3 m is very little. Certainly there will be a gap, but probably rather a very small one, our intuition says.

.

Method

The radius of the extended band results from the radius of the earth r + the distance D between earth and band. Multiplied by 2π you get the perimeter of the floating band. The perimeter of the floating band is also defined by the perimeter of the earth + 3 m.

Let’s put these two insights into an equation and adjust it according to the distance D:

Solution

According to this, the distance D between tape and earth is 0.47746482927568600730665129011754 m, i.e. approx. 0.48 m. 0.48 m is not a problematic distance according to ISO 13857 and yet much more than usually assumed.

But is it a question of the radius of the sphere? No, the radius of the earth, or of the considered sphere is no longer part of the calculation and therefore has no influence on the distance of the band to the sphere. So if such a band is placed around a tennis ball and supplemented by 3 m, the distance is exactly the same. Similarly, a band would be placed around the sun and supplemented accordingly.

Die Illustration zu diesem Paradoxon entstand im Corel DESIGNER
  • Kugel
    Kreis mit angepasstem radialen Farbverlauf über das Andockfenster Eigenschaften und dem Hilfsmittel Interaktive Füllung, direkt am Objekt
  • Band
    Kreis projiziert in der Ansicht von oben, extrudiert und Extrusionsgruppe aufgelöst, um ein Teil vor und ein Teil hinter der Kugel zu positionieren, Eigenschaften angepasst
  • Bemaßung
    Bemaßungshilfsmittel, teilweise aufgelöst und angepasst, Bemaßungshilfslinie der eigefügten 3 m durch kopieren des Kurvensegmentes des Bandes erstellt, „+3m“ mit dem Hilfsmittel Hülle frei an die Perspektive angepasst
  • Formel
    Formeleditor
The illustration of this paradox was created in Corel DESIGNER
  • sphere
    Circle with adjusted radial gradient via the Properties docker and the tool Interactive filling, directly on the object
  • Band
    Circle projected in top view, extruded and extrusion group resolved to place one part in front and one part behind the sphere, properties adjusted
  • Dimensioning
    Dimension tool, partially resolved and adjusted, dimension auxiliary line of the inserted 3 m created by copying the curve segment of the band, “+3m” freely adjusted to the perspective with the tool Envelope.
  • Formula
    Equation Editor
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