Duchamp’s intellect is laid bare in the sections of A l’infinitif on “Perspective” and the “Continuum.” In these notes Duchamp was attempting to work out the mechanics of portraying the fourth dimension […]

Henderson, 1983, page 122

The "typotranslators" of the White Box (Fig. 9), Richard Hamilton and Ecke Bonk, wrote in an introductory text (An unknown object of four dimensions) that the White and the Green Box relationship was not understood when the White box first appeared (32 years after). The distinction between them is small because all the notes were written during the same period, but according to them, it took some time to appreciate that.
Both collections of notes concern the same quest: to evade vanity engaging in an art of rigour and ideas. Nevertheless À l'infinitif records the evolution of the theory behind the forms of the Large Glass and that is probably why most of the references to Jouffret and Poincaré are in this set of notes:
[…] also, for the continuum in 4 dimensions: Poincaré’s explication, by n dimensional continuums […]

The shadow cast by a figure in 4 dimensions: on our space is a shadow in 3 dimensions (see Jouffret Geom. in 4 dim. Page 186, last 3 lines)”

Duchamp, White Box, page 77 and 81

Duchamp is revealed in a quite different way from his traditional image which was much more ironical and humorous.
Although Duchamp said in many occasions that he drawn his conclusions about the fourth dimension from Lobachesky, Reimann, Jouffret and Poincaré he denies it when interviewed in Cabanne’s:

Cabanne: On this scientific side, you have considerable knowledge…
Duchamp: Very little. I never was the scientific type.
Cabanne: So little? Your mathematical abilities are astonishing, espe¬cial¬ly since you didn’t have a scientific upbringing.
Duchamp: No, not at all. What we were interested in at the time was the fourth dimension. In the “Green Box” there are heaps of notes on the fourth dimension.

Cabanne, 1967, page 39

In the continuation of his interview he mentions fake mathematician’s and irony experts such as Princet from the cubist circle instead of true scientific influences. With the White Box analysis these ironies would become increasingly obvious.
The white box is about the artist's inspirations, aspirations, methods, references, technical and mind developing activities. Instead of exploring the fourth dimension in a strictly poetical and figurative sense (as he did in the Green Box), Duchamp writes down his individual and personal mathematical viewpoints on the 4th dimension:

“3 lines intersecting don’t determine a sp. Therefore in a space 4 lines intersecting do not determine a hyperspace”.

Duchamp, 1966, White Box (page 67)

Duchamp doesn’t explore a mathematical and scientific language because he loves science, on the contrary, he usually tries to discredit it. By using the appearance of rigour and showing his personal interpretation of the fourth dimension he achieves an irony only accessible to people that question scientific language, goals and emotional limitations:

“A finite 4 dim continuum is thus generated by a finite 3 dim. Continuum rotating (here the word loses its physical meaning)…”

Duchamp, 1966, White Box (page72)8