Using Excel to Expose pi and Discover 'pi'

These 40 slides show how Excel is used to examine arithmetic operations with the historic definition of pi=22/7 and the algebraic definition of pi=c/2r.
The outcome is a 'digital pi' of 11/7 and c/r with major implications for 'Digital Integer Mathematics' and 'On-Screen Measuring' as new domains.

The innovative concept is 'Digital Numbers' as the basis for patterns in prime numbers - the most fundamental integers - published as "Prime Numbers in Amazing Colour Patterns" on https://primenumbers.store

This experimentation with Excel as a 'visualisation tool' led to the discovery of the contributors to the 'entanglement' of pi as a letter that generalises numbers and their calculations.

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2. π=c/2r Exposed
&
‘π’=c/rDiscovered
Excel
for Visualising Number Patterns
and
Cell-by-Cell Calculations
23/07/2024
© 2024 Copyright Sabine Kurjo McNeill -Expert Advice by Dr Wolf Siegert
2

3. 3
3.1415926535897932384626433832
79502884197169399375105820974
94459230781640628620899862803
482534211706…
Here is why:
1.the historic definition of 22/7 was replaced by the circumference / diameter ratio c/2r, which has been
translated into mathematical software, without ‘digitally inspired’ thinking, or software functionality
designed for on-screen visualisations, rather than paper-based illustrations;
2.since πis fundamental for science & engineering, and essential for measuring all sorts of phenomena, it
needs to be as digitally correct and accurate as possible;
3.my new ‘Cell-by-Cell Definition’ with its digitally enhanced precision is based on my experience as a
systems analyst and software diagnostician at CERN; and I know that a solution to the enigma that π
presents has far reaching potential for future versions of Excel and Open Source libraries, as well as new
software for mathematics, data science and specific data domains and applications.
Why is my new ‘π’important for our Digital Age?
Historic
Definition
π=
c/2r
Algebraic
Definition
'π'
‘Cell-by-Cell’
Definition

4. A Numerical Formula for π, ‘π’and√2
π=22/7
320 √2= 144π
5×2
6
√2= 2
4
×3
2
π
‘π’ = 11/7
23/07/2024
© 2024 Copyright Sabine Kurjo McNeill –
Expert AdvicebyDrWolf Siegert
4
160 √2= 72‘π’
5×2
5
√2=2
3
×3
2
‘π’

5. 23/07/2024
© 2024 Copyright Sabine Kurjo McNeill –
Expert Advice by Dr Wolf Siegert
5
1 Formula:
‘π’=c/r
1 Fraction:
‘π’=11/7
The New The Difference
1 Letter: π
1 Metric:
π= 180°
3 Formulae:
π=c/2r
π=22/7
Excel’sPI()
Shape & Metric:
‘π’=360°
‘π’
π=
c/2r
‘On-Screen
Metrics’
is possible
based on
‘Digital
Integer
Mathematics’
'π'
‘Cell-by-Cell’
Definition
The Old

6. What is ‘On-Screen Metrics’?
Analogue ‘By Instrument’
1.Physical Instruments establish
measurements.
2.Instruments embody the ‘state of the art’
of their time.
3.They use Measuring Unitsderived from
•1D metre / 3D kilogram
•4D hour / seconds
based on
•electricity & magnetism
•waves & particles.
Digital ‘On-Screen’
1.Numerical Ratios establish ‘digital
boundaries’:
•of pixels on screen & of cells in Excel tables.
2.Measuring Units are geometric and visual.
•Their quantifiers are ‘attributes’.
3.Excel tables are the ‘numerical sandbox’
for:
•Circular Metrics;
•Structured Numerical Data;
•Light as Colour from Sensors;
•Digital Colour Coding of Number Patterns.
23/07/2024 © 2024 Copyright Sabine Kurjo McNeill -Expert AdvicebyDrWolf Siegert 6

7. What is ‘Digital Integer Mathematics’?
1.The Combination of:
•my discovery of those Integers called ‘Prime Numbers’, which I
published on www.primenumbers.storein ‘Amazing Colour Patterns’,
•and my concept of ‘Digital Numbers’,characterised by their Last Digit 0…9,
•in the spirit of Stephen Hawking’s book God Created the Integersin 2005,
quoting Leopold Kronecker"God made the integers, all else is the work of man.“
2.The Results of using Excel for ‘Cell-by-Cell Operations’:
•Integersas Axis Unitsfor ‘Cell-by-Cell Calculations’,such as 22 : 7& 11 : 7
and 22/7& 11/7;
•Tables ofCell-by-Cell Calculationswith Colour Coded Number Patternsfor a
future of demystifications, moving from line-basedto ‘pattern-based’ coding.
23/07/2024 © 2024 Copyright Sabine Kurjo McNeill -Expert AdvicebyDrWolf Siegert 7

8. Digital Integer Mathematics
•allows children to place Fractions with
ease outside the Number Line;
•finds Prime Numbers, missing from
Multiplication Tables, but present in
Division and other Tables of Calculations;
•quantifies ‘Numerical Ratios’ by
Pythagorean Roots to pave the way for
‘Diagonal’ and ‘CircularMetrics’;
•re-evaluates the role of √2 in the light of
my new ‘π’.
On-Screen Metrics
•is based on the Mathematics of Integer
Numbers rather than the Physicsof
Measuring Instruments;
•permits new quantifications due to
o‘Diagonal’ and ‘Digital Ratios’
othe use of Excel functions;
•allows for diagonal screen ratios to be
standardised, just as √2 became the norm
for paper formats.
23/07/2024
© 2024 Copyright Sabine Kurjo McNeill
Expert Advice by Dr Wolf Siegert
8
The Benefitsof ‘On-Screen METRICS’
and ‘Digital Integer MATHEMATICS’
'π'

9. Exposure & Discoveries
23/07/2024 © 2024 Copyright Sabine Kurjo McNeill -Expert Advice by Dr Wolf Siegert 9
For ‘On-Screen Metrics’, theπ= 22/7 FRACTIONis used as x/y Axis UNITS
•for Landscapes of Number Patterns
oleading to ‘π’with‘Metric Consistency’.
b
As ‘Method of Exposure’, Excelis used for
•Cell-by-Cell Calculationsfor Multiples & Fractions < = > 1andπ
oand VisualComparisons between
•DiagonalRatios andOrthogonalDirections
•Pythagorean Roots of2and other SumsofSquares
•RatiosasDEGREESforFractions of πand ‘π’.
Historic Definition
'π'
‘Cell-by-Cell’ Definition

10. Exposure & Discoveries
Analysisfor Exposure
1.The generalisationof Numbersby Letters
in Algebra;
2.The equalisation of arithmetic Operations by
‘operational units’ 1 ×1, 1 / 1, 1
n
, √1;
3.The difference of Calculationsbetween Excel
Tablesand AlgebraicMatrices;
4.Thecomparison between Multiples and
Fractions < = > 1;
5.Numerical‘Step-by-Step Ratios’become
Geometric Diagonals.
Conceptsfor Innovations
23/07/2024
© 2024 Copyright Sabine Kurjo McNeill -Expert AdvicebyDrWolf Siegert
10
Historic Definition
π=
c/2r
Algebraic Definition
1.Colour Coded PATTERNSof Calculations
replace Formulae for Coding SERIES;
2.Symmetric22x / 7yand 11x / 7yDivisions
reveal:
•a 7 ×11Grid for Multiples & Fractions of π& ’π’
•‘Diagonal Growth’ and ‘Orthogonal Constancy’
•Orthogonal Factorisation of Diagonal Quotients
3.Excelidentifies 9as Multiple of ‘Proportional
Degrees’ in Horizontal : Vertical Ratiosfor
•SQRT (Sumsq) of Diagonal Lengths
•DEGREES (Atan (x/y) for Diagonal Angles.

11. π=c/2r
π=22/7
Excel’sPI()
23/07/2024 © 2024 Copyright Sabine Kurjo McNeill -Expert AdvicebyDrWolf Siegert 11
‘π’=c/rand‘π’=11/7 redress the issue:
‘METRICCONSISTENCY’ensues for Fractions of π,
Numerical Ratios, Roots&Circular 360°Metrics.
In a Nutshell:
πis the ‘entanglement’ of:
•generalisingnumbers by LETTERS,
•inverting divisions into MULTIPLICATIONS,
•equalising‘UNITSof operation’: 1=1
n
=1/1=1×1,
•and repeatingcalculations in SERIES.

12. What are ‘Cell-by-Cell Operations’?
1.Symmetrical Divisionsof π=22/7 and ‘π’ =11/7
•for comparing DiagonalMultiplesand Fractionsof π;
2.Symmetrical Factorisation of π= 22/7 and ‘π’ =11/7
•for identifying the Factorsof their Multiples and Fractions;
3.Pythagorean Rootsof ‘Numerical Axis Ratios’ x : y
•for quantifying Diagonal Lengths;
4.DEGREES (Atan)
•for quantifying Anglesin degrees;
5.Digital Colour Coding
•for highlighting Number Patterns of Samenessand Differentness.
23/07/2024
© 2024 Copyright Sabine Kurjo McNeill –
Expert AdvicebyDrWolf Siegert
12

13. What ‘Cell-by-Cell Divisions’Reveal
1.Arithmetic Divisions of 22x / 7y reveal Integers in Positions that
form a 7 ×11Grid.
•These Integers are Quotients that are either Multiples or Fractions of
π=22/7,i.e. they are >= <π.
•Their ‘Cross-Line Positions’ form Horizontal, Vertical and Diagonal Series.
•In particular, Diagonal Series are formed by π, 2πand 4π.
•11appears in 7 ×4Rectangles.
2.The Table of Divisions also reveals Multiples of √2 in orthogonal and
diagonal positions in regular intervals.
•10√2 appears in 9 ×4Rectangles.
23/07/2024
© 2024 Copyright Sabine Kurjo McNeill –
Expert AdvicebyDrWolf Siegert
13
Sheet: 2r Grid

14. 23/07/2024
© 2024 Copyright Sabine Kurjo McNeill -Expert AdvicebyDrWolf Siegert
14
Diagonals
Red: πor1H : 2V
Orange:
2πor1H : 1V
Green:
4π or 2H : 1V
Sheet: 2r Grid
CELL-by-CELL DIVISIONS
NUMERICAL FRACTIONS 22x/7y investigate the VALUEof the LETTER π
Fractions of π < 1
DIAGONAL SERIES
revealMultiples
& Fractionsof
Decimal Factors
and of π.
11is at the Corner of
7 x 4Rectangles

15. 23/07/2024 © 2024 Copyright Sabine Kurjo McNeill -Expert AdvicebyDrWolf Siegert 15
Diagonal Series
½π:1H : 2V
or 2V : 1H
‘π’as 1H : 1V
in4 Directions
2π:2H : 1V
or 1V : 2H
Sheet: r Grid
Fractions of ‘π’< 1
11is at the Corner
of 7 x 2Rectangles
CELL-by-CELL DIVISIONS
NUMERICAL FRACTIONS 11x/7y investigate the VALUEof the LETTER ‘π’
The 7 ×11 Grid
shows proportional
Factorsof ‘π’in
Cross-Line Positions.

16. π=22/7 vs‘π’=11/7
Cell-by-Cell Divisions Compared
23/07/2024 © 2024 Copyright Sabine Kurjo McNeill -Expert AdvicebyDrWolf Siegert 16
Same Grid: Different Axes, Diagonals & Areas < 1
1H : 1V:2πvsπ; 1H : 2V: πvs½π; 2H : 1V: 4πvs 2π

17. 23/07/2024
17
Cell-by-Cell FACTORISATIONS and DIVISIONS of 22x/7y
Below: MULTIPLES& FRACTIONSof π
Rose: ½π or4V : 1HRed: πor 1H : 2V
Orange: 2πor 1H : 1V Yellow:3πor3H : 2V
Green: 4πor 2H : 1V
Sheet: 2r Factors
Above Axes: DIAGONALSof FACTORS
VERTICALLY PRESENT in QUOTIENTS below
Rose: 1/2Red:1Orange: 2
Yellow: 3Green:4
π = 3.14asFACTOR
484 = 2
2
×11
2
x/y AXES for Multiples of 22and of 7

18. 23/07/2024 18
Below: MULTIPLES& FRACTIONSof ‘π‘
Rose: ¼π or 1H : 4V
Green: ½π or 2V : 1H
Red: πor 1H : 2V
Yellow:3/2π
Orange: 2πor 2H : 1V
Sheet: r Factors
Above Axes: DIAGONALSof FACTORS
Rose: ½; Red:1; Pink: 3/2
Orange: 2Yellow: 3
Green:4
π = 3.14and π/2 = 1.57asFACTORS
242 = 2×11
2
484 = 2
2
×11
2
Cell-by-Cell FACTORISATIONS and DIVISIONS of 11x/7y

19. π=22/7 vs‘π’=11/7
in Cell-by-Cell Factorisations & Divisions
23/07/2024 © 2024 Copyright Sabine Kurjo McNeill -Expert AdvicebyDrWolf Siegert 19
SameValues: DifferentDiagonals& Factors
1H : 1V= 2πvs1H : 1V= π;π=3.14 &π/2=1.57

20. 23/07/2024
© 2024 Copyright Sabine Kurjo McNeill -Expert Advice by Dr
Wolf Siegert
20
Cell-by-Cell ROOTS quantify 22 : 7RATIOSas ‘STEP DIAGONALS’
DIAGONALS
of
MULTIPLESof π
Rose:π/2or 1H : 4V
Red: πor 1H : 2V
Orange: 2πor 1H : 1V
Green: 4π or 2H : 1V
Blue: 8π or4H : 1V
Sheet: 2r Py
RATIOSare
Horizontal : Vertical
‘STEP DIAGONALS’
in Central 4-fold
and Diagonal
8-fold Symmetry

21. 23/07/2024 © 2024 Copyright Sabine KurjoMcNeill -Expert AdvicebyDrWolf Siegert 21
Cell-by-Cell ROOTS quantify11 : 7RATIOS
as Multiples of‘Diagonal Units’ for Radii of Proportional Circles
DIAGONALS
of
MULTIPLES
of ‘π‘
π/4or 1H : 4V
π/2or 1H : 2V
πor 1H : 1V
2πor 2H : 1V
4π or4H : 1V
Sheet: r Py
FACTORS
multiply RATIOS
along DIAGONALS

22. 23/07/2024 22
Pythagorean Roots: SameDiagonals, DifferentValues
1 : 4=√17= ½π
1: 2= √5= π
1: 1=√2= 2π
2: 1= √5 = 4π
4: 1 = √17 = 8π
π=22/ 7 ‘π’=11/ 7
Diagonal LENGTHS
as ROOTS, RATIOS,
FRACTIONS & MULTIPLES of
3.14are consistent with ‘π’
1 : 4 = 1/4π
1 : 2 =½π
1 : 1 = π
2 : 1 =2π
4 : 1 = 4π

23. 23/07/2024
1.4-foldOrthogonal Symmetryof
Circular DIVISIONSinto Quadrants;
2.8-foldDiagonal Symmetryof
Horizontal : VerticalRATIOS
intoOctants;
3.theTRANSFORMATIONof‘Digital Ratios’
intoDEGREESviaExcelFunctions;
4.the DiagonalCONSTANCYof Degrees;
5.the Circular GROWTHof Angles
in Steps of 9, from 9 to 351;
6.the ‘digitally proportional’ role
of 11 :8for 22: 16, 33: 32or
n×(10
0
+10
1
) :2×2
3
.
Sheet: 9 Angles
Cell-by-Cell ANGLESin Excel DEGREES illustrate:

24. 23/07/2024 24
Cell-by-Cell ‘ROOT UNITS’ for Dividing Circles
SYMMETRICAL RATIOS
for MULTIPLES of 9DEGREES & ROOT UNITS
6: 1= √37= 9°81°99°171°189°261°279°351°
3: 1 = √10= 18°72°108°162°198°252°288°342°
2: 1= √5= 27°63°117°153°207°243°297°333°
11 : 8= √185=36°54°126°144°216°234°306°324°
1 : 1= √2=45°135°225°315°
DIAGONAL RADII are
MULTIPLES of ‘ROOT UNITS’
Sheet: 9 Roots
‘ROOT UNITS’ are Pythagorean Roots
of Sums of Squares of ‘DIGITAL RATIOS’

25. 23/07/2024 25Sheet: 9 Diagonals
Cell-by-Cell ‘ROOT RATIOS’ for ‘Diagonal Repetition’
1 : 1= √2 determines
Diagonal ‘VISUALISATION UNITS’
as n ×√2.
Rootsare derived from Ratios, visualised as
dotted UNITS and straight REPETITIONS.
Anglesare derived from Ratios as Quotients,
visualised as diagonal LINES and CELLS.
√2 is the Smallest Root Unit
CENTRAL 1 : 1DIAGONALS
are INTEGER Multiples of √2

26. © 2024 Copyright Sabine Kurjo McNeill -Expert AdvicebyDrWolf Siegert
26
DEGREES & FRACTIONS RATIOS ROOTS
Sheets 9 Angles, 9 Roots & 9 Diagonals summarised

27. 23/07/2024 27
22:7and 11:7RATIOSfor ANGLES in Excel DEGREES
ATAN translates RATIOSinto Radians.
DEGREES translates them into degrees.
Excel DEGREES (ATAN (22/7) ) =72.35° Excel DEGREES (ATAN (11/7) ) =57.53°
Diagonal ANGLES
are STEP : STEP RATIOS
between Cells
Radians [57.296°] are FRACTIONS of π
in terms of SEMICIRCLE or 180 DEGREES
Sheet: r AtanSheet: 2r Atan
The
degrees are
‘digitally
coloured’
according
to their
‘Last Digit’
to illustrate
the
differences.

28. 23/07/2024 28
10√2 = 14.14 =
4.5π
20√2 = 28.27 =
9π
40√2 = 56.55 =
18 π
80√2 = 113.1 =
36π
120√2 = 169.6
=54π
160√2 = 226.2
=72π
320√2 = 452.4
= 144π
Sheet: 2r Grid Rt2
Cell-by-Cell DIVISIONSreveal Multiplesof √2in ‘Ortho-Diagonal’ Positions
Multiplesof √2meet Multiplesof π,from10to320√2: 5×2
6
√2= 2
4
×3
2
π

29. 23/07/2024 29
10√2 = 14.14
= 10‘π‘
20√2 = 28.27
=20‘π‘
40√2 = 56.55
= 40 ‘π‘
80√2 = 113.1
= 36‘π‘
120√2 = 169.6
=54‘π‘
160√2 = 226.2
= 72‘π’
Sheet: r Grid Rt2
Cell-by-Cell DIVISIONSreveal Multiplesof √2in ‘Ortho-Diagonal’ Positions
Multiplesof √2meet Multiplesof ‘π‘,from10to160 √2: 5×2
5
√2= 2
3
×3
2
‘π’

30. 23/07/2024 30
Multiples of√2in ‘Ortho-Diagonal’ Positions
π=22/ 7 =>1 : 1=6.28… ‘π’=11/ 7 =>1 : 1= 3.14…
320 π= 144√2 and 10√2 in 9 ×4Rectangles 160‘π’=72√2 & 10√2 in 9 ×2Rectangles
Factor 9for π& ‘π’ Factor 10for √2
9is for Degreesin Circleswhat10is for Diagonals in Squares:
the ‘Unit Factor’for Multiplications.
Sheet: 2r Grid Rt2 Sheet: r Grid Rt2

31. π/√2&the‘DIGITALFABRIC’ of DECIMALS> = < 1
23/07/2024 31
DIAGONALDECIMALS: 1.11, 2.22, 3.33, 4.44, 5.55,6.66…
Sheet: 2r by Rt2 Sheet: r by Rt2

32. Once Again:
Same Diagonals, Different Values, but same PRINCIPLE: DIAGONAL RATIOS!
23/07/2024 32
8.89 for 2: 1, 6.66for 3 : 2,4.44for 1 : 1,2.22for 1 : 2
and 1.11 for1 : 4
Sheet: 2r by Rt2 Sheet: r by Rt2
8.89 for 4: 1, 6.66for 3 : 1,4.44for 2: 1,2.22for 1 : 1
and 1.11 for1 : 2
DIAGONAL RATIOSas DECIMAL ‘SAME DIGIT’ SEQUENCES

33. 23/07/2024 © 2024 Copyright Sabine Kurjo McNeill -Expert AdvicebyDrWolf Siegert 33
‘Visi-Metric’ Principles at the Root of ‘π’
•Algebra vs Geometry
•Lettersvs Numbers
•‘Dimensional Metrics’
•2DCircumference vs 1DDiameter
•Symmetry
•4 Quadrants: ←→and ↑↓
•Metrics vs Visualisation
•Whole Circumference & Semicircle
•Quotientsfor Ratios
•Formulae vs Shapes
•Series vs Patterns
•Semicircle vs Circle
•Diameter vs Radius
•8 Octantsin 4Quadrants
•1 : 1for ↖↗↙↘and < = > 1
•‘Visual Metrics’
•180vs 360Degrees
•Pythagorean Rootsfor Ratios
System Analysis Principal Innovations

34. ‘Number Patterns’ at the Root of ‘π’
•Ratios = Quotients
oDivisions are Diagonal
•Pythagorean Roots
oTrigonometricRatios
determine the Lengths of Diagonals
•Proportionality
o1 : 1 = √2for 45°
•Colour Coding for Sameness
23/07/2024
© 2024 Copyright Sabine Kurjo McNeill -Expert AdvicebyDrWolf Siegert
34
•Ratios=PYTHAGOREANROOTS
oDiagonals are ‘Proportionality
Factors’
•‘Diagonal ROOTUNITS’
o1 : 1for ‘Central’ Multiples of √2
on : m= m : nfor Mirrored Rectangles
•Proportional DEGREES
oMultiples of 9
oOctants for < = > 1[45°] Comparisons
Physical Measuring Units Screen-Based Measuring Units

35. When Excel calculates Degrees from Atan (Quotients)
23/07/2024 35
Degrees (Atan (22/7) = 72.35° Degrees ( Atan (11/7) = 57.53°
28, 46, 58, 66, 72, 77, 81, 84, 87
Sheet 2r Atan D2 Sheet r Atan D2
15, 28, 38, 49, 58, 65, 72, 78, 84
Same Diagonals, different Values

36. For Every ‘π-Fraction’ its Diagonal of Ratios in Degrees
23/07/2024 © 2024 Copyright Sabine Kurjo McNeill -Expert AdvicebyDrWolf Siegert 36
π=22x /7y ‘π’=11x/ 7y
Sheet 2r Atan Sheet r Atan

37. In Summary: π vs‘π’
•π is a remarkably consistent ‘entanglement’ of
algebraic operations, arithmetic calculations and geometric relationships;
ogoing round in circles instead of swinging from semicircle to semicircle;
•‘π’ pays tribute to the value of numericalrelationships rather than the
advantage of algebraic generalisations by letter;
ovalue cannot come from letters, only numbers;
•‘π’ is the result of the SYMBOLISMof digitsover numbers and programming
over mathematics;
oit’s not about decimals behind the point; it’s about INTEGERS;
•NUMBERPATTERNSare stronger than any other ‘tool of conviction’ or ‘method
of proof’;
oneither numbers nor their patterns lie; only words as interpretations.
23/07/2024
© 2024 Copyright Sabine Kurjo McNeill –
Expert AdvicebyDrWolf Siegert
37

38. Ask Uncle π
1.On what grounds
does a Greek letter get attributed
with Decimals when Integers can help?
2.Who will decide how ‘π‘
can exist in the ‘digital shadow’ of π?
23/07/2024
© 2024 Copyright Sabine Kurjo McNeill –
Expert AdvicebyDrWolf Siegert
38

39. π vs‘π’: Essential Differences
23/07/2024 39
Radius
11/7
360°
Degrees
(Atan(11/7))
= 57.529
‘π’/√2=1.11
Diameter
22/7
180°
Degrees
(Atan(22/7))
= 72.35
π/√2=2.22
57.296 is the
conventional constant
for a ‘radian’.

40. Feedback on LinkedIn please!
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