The document discusses patterns that can be found in Pascal's triangle. It notes that Pascal himself remarked on the fertility of properties to be found in the triangle. It provides two examples of patterns, where multiples of 5 and 7 are picked out in the triangle. The reader is also prompted to explore if the coefficients of trinomial expansion can be represented in a pattern.

1. ΠΡΟΤΥΠΟ ΠΕΙΡΑΜΑΤΙΚΟ Γ/ΣΙΟ
ΙΩΝΙΔΕΙΟΥ ΣΧΟΛΗΣ ΠΕΙΡΑΙΑ
ΥΠΕΥΘΥΝΟΙ ΚΑΘΗΓΗΤΕΣ
Αρχοντάκη Α. , Παγιαυλάς Ι. ,
Παρασκευουλάκου Α.
More Pascal Patterns
Των μαθητών του Γ4:
Γαληνός Δ. , Τζάκης Γ.
Βράνος Η. (Α) , Βράνος Η. (Γ)
ΣΧΟΛΙΚΟ ΕΤΟΣ 2014-2015 ΠΕΙΡΑΙΑΣ

2. MORE PASCAL PATTERNS
The features of this elegant number pattern which we have briefly described are
sufficient to show that aesthetic pleasure is to be derived from arithmetical patterns,
but we have only touched the fringe of the possibilities. Pascal himself remarked in his
treatise on the subject how fertile in properties this triangle proved to be.
Fig. 10.1
https://www.google.gr/search?q=pascal%27s+triangle&espv=2&biw=1680&bih=925&source=lnms&tbm=isch&sa=X&ei=
6eCuVJKGOdStaaPYgKgD&ved=0CAYQ_AUoAQ#tbm=isch&q=pascal%27s+triangle+patterns&revid=272279997&imgdii=_
&imgrc=2h5rEBed3g95nM%253A%3BNSpZ18qYxhdhcM%3Bhttp%253A%252F%252Fwww-
math.ucdenver.edu%252F~wcherowi%252Fpt5.gif%3Bhttp%253A%252F%252Fwww-
math.ucdenver.edu%252F~wcherowi%252Fjcorn5.html%3B602%3B528

3. MORE PASCAL PATTERNS
The reader may care to search for himself. Can the coefficients of trinomial expansion
(a+b+c)ᵐ be represented in to a pattern?
Before we take leave of this topic we would draw attention to the patterns based on
Pascal’s triangle in figures 10.1 and 10.2. Figure 10.1 shows the relative positions in
Pascal’s triangle of all the multiples of 5. In figure 10.2, multiples of 7 are picked out. The
results ar an unexpected as in the case of the magic squares.
Fig. 10.2
https://www.google.gr/search?q=pascal%27s+triangle&espv=2&biw=1680&bih=925&source=lnms&tbm=isch&sa=X&ei=
6eCuVJKGOdStaaPYgKgD&ved=0CAYQ_AUoAQ#tbm=isch&q=pascal%27s+triangle+patterns&revid=272279997&imgdii=_
&imgrc=2h5rEBed3g95nM%253A%3BNSpZ18qYxhdhcM%3Bhttp%253A%252F%252Fwww-
math.ucdenver.edu%252F~wcherowi%252Fpt5.gif%3Bhttp%253A%252F%252Fwww-
math.ucdenver.edu%252F~wcherowi%252Fjcorn5.html%3B602%3B528

5. The Grand Egyptian Museum
The site for the Grand Egyptian Museum is located at the edge of the first desert plateau
between the pyramids and Cairo. It is defined by a 50m level difference, created as the Nile
carves its way through the desert to the Mediterranean, a geological condition that has
shaped Egypt for over 3,000 years.

6. The Grand Egyptian Museum
The pyramids, funerary monuments, are located in the desert on the plateau 2km from
the museum site, while the site for the museum is located both in the valley and on the
plateau.

7. The Grand Egyptian Museum
The design of the museum utilizes the level difference to construct a new ‘edge’ to the
plateau, a surface defined by a veil of translucent stone that transforms from day to
night. The museum exists between the level of the Nile Valley and the plateau, never
extending above the plateau.

8. The Grand Egyptian Museum
The approach to the museum is a series of layers, whereby the visitor moves through a
monumental forecourt, a shaded entrance area and a grand staircase that ascends to
plateau level, the level at which the galleries are located where for the first time the
visitor sees the pyramids from within the museum.

9. The Grand Egyptian Museum
The museum is envisaged as a cultural complex of activities devoted to Egyptology and
will contain 24,000m² of permanent exhibition space, almost 4 football fields in size, a
children’s museum, conference and education facilities, a large conservation centre and
extensive gardens on the 50hA site. The collections of the museum include the
Tutankhamen collection, that is currently housed in the Egyptian Museum in Cairo, and
the Solar Boat that is now housed beside the pyramids.

10. REFERENCES
http://en.wikipedia.org/wiki/Pascal%27s_triangle
https://www.google.gr/search?q=pascal%27s+triangle&espv=2&
biw=1680&bih=925&source=lnms&tbm=isch&sa=X&ei=6eCuVJK
GOdStaaPYgKgD&ved=0CAYQ_AUoAQ#tbm=isch&q=pascal%27s+
triangle+patterns&revid=272279997&imgdii=_&imgrc=2h5rEBed
3g95nM%253A%3BNSpZ18qYxhdhcM%3Bhttp%253A%252F%25
2Fwww-
math.ucdenver.edu%252F~wcherowi%252Fpt5.gif%3Bhttp%253
A%252F%252Fwww-
math.ucdenver.edu%252F~wcherowi%252Fjcorn5.html%3B602%
3B528
http://www.hparc.com/work/the-grand-egyptian-museum/
H.E. HUNTLEY, The divine proportion,
Dover Publications, N.York 1970
Α.ΑΛΙΜΠΙΝΙΣΗΣ, Σ.ΓΡΗΓΟΡΙΑΔΗΣ, Ε.ΕΥΣΤΑΘΟΠΟΥΛΟΣ,
Ν.ΚΛΑΟΥΔΑΤΟΣ, Σ.ΠΑΠΑΣΤΑΥΡΙΔΗΣ, Α.ΣΒΕΡΚΟΣ ,
ΜΑΘΗΜΑΤΙΚΑ Γ΄ ΓΥΜΝΑΣΙΟΥ . ΟΕΔΒ,ΑΘΗΝΑ 2004