Shapes Based On Parametric Equations: Epicycloids

Epicycloid is a parametric geometric curve obtained by tracing the path of a chosen point on the circumference of a circle (outside of it). Variety of epicycloid curves find applications in mechanical and construction engineering, e.g. construction of gears, cams, valves, pendulum clocks, robotic actuators, and machine/structures design in general. Contribute also to making designs of arts and animations.

In this context, it is useful to know how to generate epicycloid curves. In this post I'm presenting some of the curves along with the Excel VBA macro for creating this kind of curves. You can try to use it for your own creations. Just copy it to one of the modules in your workbook sheets and experiment with different settings and parameters.

Here are some examples of charts with epicycloid curves, single and double plots. 

 

Curves Based On Equations: Lemniscate of Bernoulli

The Lemniscate curve is based on specific algebraic polynomial equation. It has a shape similar to the numeral 8 and to the infinity symbol ∞ (called also a ribbon). You can create it with Excel VBA code and use it as one of the elements of graphic design.

Here are two examples of the curve created with the macro code listed below them.

Excel Comments: Setting Font Format

Default font format in Excel Comments may look like this:

If you want to change some of the font settings (i.e. the Windows font settings), you can either try to adjust font format in "Display settings" of Windows or use VBA code listed below. The second option is better, because it doesn't change font settings in all other Windows applications. Remember that you must make your Comments visible. If necessary, right-click in the cell and click on Show/Hide Comments option.

This is the event driven macro. To use it, you need to:

• open the Visual Basic for Applications (VBA) editor in Excel (use the shortcut ALT+F11 on your keyboard)
  
• click on the View in the top menu and choose Project Explorer option
• in your VBAProject double-click on Sheet1 and paste the macro code in the window
  
• close the VBA editor and return to your Excel worksheet

Excel Shapes: Positioning, Formatting and Layout In Worksheets

Excel Shapes are frequently used to enhance visual appeal of worksheets as well as to convey information. It's relatively easy to work with the Shapes. You can insert, resize, rotate, format, and position them within the worksheet, according to your needs.

This post is about special case of positioning and manipulating of Shapes. Namely, I'll show the way of inserting and moving them so that they are located/centred precisely at the crosspoint of 4 surrounding it cells. Furthermore, in this case I'll use a worksheet with squared cells. If you need to know how to make cells square please refer to one of my previous posts (Make cells SQUARE) .

Here are just two examples showing what I'm talking about:

 

 

These are the formatted Shapes called "Donut" and "Gear with nine teeth". They are placed exactly at the point where the four cells meet together. It's not that easy to do that by hand, so I've created the following Excel macro, to handle this task swiftly:

Prime Numbers: Their Importance In Computational Security and Other Applications

Prime numbers primarily constitute the basis of cryptography (hidden writing) and cybersecurity, i.e. secure transmission and storage of digital data. They are used in mathematical algorithms that enable efficient encryption (converting into code) and decryption (converting to plain text) of data.

And everything starts with generating and selection of just two prime numbers. However, the generated primes must be really very high numbers for the purpose. The higher they are the better becomes security/privacy of communication between parties.

Beside that, the whole concept of cryptography relies also on the complexity of applied algorithms (ciphers). And these are still based on prime numbers and are evolving all the time.

Cryptocurrencies, such as Bitcoin, are also created with with cryptographic algorithms and rely on prime numbers for ensuring secure transactions and protecting private information.

Primes are used as well in mechanical design, machinery and engineering structures. Choosing primes for e.g. the number of gear tooth, blades, or similar structural elements, helps to ensure reduction of the wear of materials, reduction or better distribution of vibrations, eliminating unwanted resonances.

In machine design - uniform gear wear, reduced noise or desired gear ratios can be achieved by choosing the tooth counts to be co-primes (relatively prime), i.e. 7 & 12, 13 & 21, or 7 & 9 & 12. This is an example of gear with co-primed  numbers of cogs: 7 & 12.

 

 

In electrical engineering - prime numbers are used in the design of power distribution system to improve the efficiency of the system.

There are also dozens of important uses for prime numbers in many sciences. So, if you are really interested and want to be better educated in the subject of primes and related to them algorithms it's worth to familiarize yourself with one of the most stimulating recent publications by Gary William Croft: "The Prime Spiral Sieve". It can be found at https://www.primesdemystified.com .

Prime Numbers: Presentation of Distribution of Primes with Waterfall Charts

One way of effective visual presentation of prime numbers is using Excel charts. And the Waterfall type of chart seems to be quite useful. It shows a running total as values are added to your data set; in this case - as consecutive primes are added. The running totals can be shown on the vertical axis as well as on the chart itself (for each subsequent prime value).

Here are some examples of this kind of visualization of running totals of primes. You'll need to enlarge the charts or use great size screen area in order to see the charts in detail.

Prime Numbers: How to Find the Highest Prime Number Below Given Limit

If you need to find/generate the highest prime number not exceeding some given limit, like e.g. 100,000, using the following Excel VBA macro is the appropriate and easy to use tool; obviously, within the Excel program specifications.

Just copy and paste the code into one of the modules of your workbook. Some explanatory notes are included within the macro.

Sub CalcTopPrimeNumberBelowGivenLimit()
'Max prime <1,000,000: 999,983
'Max prime <10,000,000: 9,999,991
'Max prime <100,000,000: 99,999,989
'Max prime <1,000,000,000: 999,999,937
'For max. integer handled by my computer (=1,068,699,999) the highest prime is: 1,068,699,979
    Dim n As Long
    Dim primes() As Boolean
    Dim i As Long, j As Long
    Dim tPrime As Long
    'Set the maximum limit for prime number generation
    n = InputBox("Enter your max. limit for the prime generation:", "Set the max. limit", 199999999)
    'Initialize the array to assume all numbers are prime
    ReDim primes(2 To n)
    For i = 2 To n
        primes(i) = True
    Next i
    'Apply the Sieve of Eratosthenes algorithm (it finds all primes up to a given limit)
    For i = 2 To Sqr(n)
        If primes(i) Then
            For j = i ^ 2 To n Step i
                primes(j) = False
            Next j
        End If
    Next i
    'Find the highest prime number
    For i = n To 2 Step -1
        If primes(i) Then
            tPrime = i
            Exit For
        End If
    Next i
    'Display the result
    MsgBox "The highest prime number below " & n & " is " & tPrime
End Sub